What do I do when I'm alone in a lane?

Don't die.

First you should always consider these 2 things...

Should I be alone in lane? (is there a team fight, objective, etc...)

Am I safe? (Are there enemies pushing, do you know where they are?)

If the top two things are in your favor work your way down this list but before moving down always repeat the top two steps.

Can you safely clear the minions?

Should I be alone in lane? (is there a team fight, objective, etc...)

Am I safe? (Are there enemies pushing, do you know where they are?)

Can you then safely push a base?

Should I be alone in lane? (is there a team fight, objective, etc...)

Am I safe? (Are there enemies pushing, do you know where they are?)

If you manage to down a base and the next wave a minions are cleared and the top 2 things still do not apply take a camp or join with your team to push another base maybe.

You kinda answered your question :)
If you're safe to push - it's always good to push. I usually consider own minions health status and estimation of when enemy will return to lane. The goal is to gain all exp possible and to deny all exp possible. If I expect someone to return really soon, I usually soak more passively, letting my minions to die and enemy exp to be wasted. Otherwise, I clear the wave and let my minions to die under towers. They die faster, and take out towers ammo, but it takes some time to reach the towers.
It's rarely a good idea to commit to pushing structures while alone in lane, IMO, because when you are 100% sure that all enemies are far, probably your team needs you there, too.

If youre on a small map clear the lane and rotate if you have good waveclear and any gank potential.

If you know they are roaming then DO NOT PUSH. Assume that diablo and tyrande are in the bush and position accordingly. Dont worry about your turrets using ammo or taking some dmg, thats peanuts compared to dieing and losing soak. If they are pushing then ofc ping.

If all enemies are on map then clear wave and stay if the map is big and/or you dont have much gank potential.

EDIT: Wall of text because there are a lot of things to consider with this simple question.

TL;DR -- watch your minimap, soak xp safely unless you know it's safe to push your lane.

Top priority when alone in a lane: soak XP. Make sure you are in range to get all the xp from the minions. Now your positioning when soaking depends on, essentially, enemy comp and presence on the minimap.

First question should be: does enemy team have a gank comp? Tyrande + mura/etc/diablo/arthas? Or Kerrigan, nova, zera? Even KT/Uther can be a gank squad. If the enemy team has gank heroes (and in this meta they're super popular), then you need to keep an eye out for them on your map at all times early game.

Let's say you're top. You see Diablo, Tyrande, and KT are bot. If you're laning against someone top, you should be trying to zone them away from XP if you can, scaring them back to their gate while their minions kill your minions in the middle of the lane. This means you DO NOT kill their minions. If you kill the enemy minions, great you get a globe, but then those minions walk towards the tower and the enemy player can safely soak their xp from behind the gate. You did minimal damage to the towers, used maybe 4 tower shots from each tower, and enemy got all their xp. Not really a win. You want to zone them away from xp (while soaking it yourself), so they lose xp. That's best case scenario.

If the enemy seems to have abandoned top lane where you are, then you want to shove your lane hard. Kill all the enemy minions, get your xp, and hopefully the friendly minions die to the towers with no one around to soak that xp. Once you've shoved your lane, your options are to mount and retreat back to your gate (if the enemy is smart they'll be coming to try to punish you for the shove) or, if you see an engagement going on and all 5 enemy heroes occupied, you can try to get a bit of free damage on the enemy tower/gate in your lane, but you have to be VERY careful here; I would not push in for damage if ANY enemy gank hero is mid lane after you shoved top.

Now if you are in lane and you only see 3 enemy heroes on minimap, assume the other 2 are setting a trap for you in a nearby bush, waiting for you to move away from your gate/tower so they can gank you. The strategy here is to simply mount up, and try to be in as safe a position as possible while soaking your xp. Hopefully, the enemy player in your lane is just shoving lane, which is good for you because it minimizes the time you need to be outside of your gate (if he shoves lane, you can retreat behind your gate and soak safely there). If he's smart, like I mentioned above, he will ignore your minions and attack you on your mount, trying to dismount you and scare you behind your gate where you'll miss xp soak. Until you see those other 2 heroes show up on your minimap, you have to let him do that... maybe dodge his advances on your mount and run back in just as the enemy minions are being last-hit to get the xp.

What's the counter to the enemy zoning you? Well, if he's zoning you he's put himself in an aggressive position, meaning if your team has a gank squad they could gank him. Again, consider the possible intelligence of your opponent. In high MMR games, I wouldn't expect an enemy to zone you from xp in a lane unless he felt safe, meaning probably his gank squad is close OR your team is showing all 4 other heroes on the map in other lanes, so he knows a gank isn't coming. In those scenarios, you can't do much, gotta play safe, and your priority is to not get caught and die. If you die solo in a lane, your team will likely miss all that xp before you rez and/or someone rotates to soak. Hopefully your team is winning other lanes as there are 4 teammates for the remaining two lanes and the enemy is only showing 3 on map, including the guy top with you. If your team is smart, though, they'll send some of your own gank heroes to try to punish the enemy who is zoning you, as he will be positioned very aggressively, but they need to be careful about their rotation because the enemy gank squad is likely very aggressively positioned to quickly punish an overextend by you top.

Ultimately your play in a solo lane depends on what information you see on your minimap about your opponents. If you don't see enemies, assume they're coming for you, particularly if you're an enticing target, like a KT or a support without escapes. This is why characters like Zagara are so good in a lane; her creep can be used to provide more vision to alert you to incoming ganks or aggressive enemy positions near you. More vision means more information which can inform your play even more.

So while the other advice in this thread, particularly "don't die" is valid and important, it's because your TOP priority is to soak xp, and your team will not be able to get a wave of xp if you die because of aggressive positioning (not to mention the enemy team gets xp from your death and a power play).

Hope that helps.

One thing you'll see a lot of pro players do is if they can safely clear a wave and nobody from the enemy team is currently there, they will push the wave into towers then roam a little hoping to dismount the opponent who wants to soak that wave

When the opposing team is not all around my team in a team fight, I watch the minimap, and what you see there alone should give you all the clues you need to answer those questions :)

Are they 4 bot and you are top? who is missing? Only Johanna? Then you can very likely feel safe to push with thrall. (insert a billion examples and counterarguments)

Minimap is love, minimap is life.

Cry

When I'm alone in lane, I watch the mini-map. Just like u/scottlohm is saying, make sure you're safe to push and make sure your team doesn't need you. Then you can stay in lane. But ALWAYS watch the mini-map for that information. Where enemies are, etc.

A little me time ?

Try to never be sitting idle doing nothing. If you're alone in a lane, clear it up and rotate to help your team where they're outnumbered. If your team is fine and you see all of the enemy team on the map, put pressure in your lane while looking every couple seconds at your mini map; if you see even one enemy missing, back off immediately.

When you are alone in lane and think/know that opponents are far away enough then clearing can actually deny more exp than waiting. A quick waveclear lets your minions push to enemy towers where they die faster than to enemy minions so they can miss out on an entire wave of exp.

Its not really a set rule. Early game when you want xp you don't wanna leave a lane unless a bigger priority appears. You might find yourself in a 1v2 which can be painful but if you're good you shouldn't die. You still wanna at least be getting that xp. Also pay attention to the map when solo laning, even more than usual. ESPECIALLY if there are also no enemies in the lane with you. Once you're out of the fog for the enemy, and they see you're milking xp where they aren't, I assure someone will come to attack you. Also create a priority system of whats more important than 1-2 minutes of xp. Usually these things are Objectives, team fights, etc. Obviously if your team has a 1-2 level lead, you can be a little more free with your time. Also try not to siege unless you have a camp preferably. Or if you do, basically you need to be waiting for the fog attack from behind the gate. You don't wanna get caught in a stunfest in their turrets just to get bursted down. And on top of all of that, every single character solo lanes differently. So it really depends who you play. I expect an assassin and a specialist to behave much differently when soloing a lane. Also on the note of rotating, I personally wouldn't unless its on a smaller map like you said, or the enemy team is essentially forcing you to. tl;dr: play it safe, watch your map, milk xp (early -mid) for late, play it safer, watch your map even harder, try to put pressure on their constructs

First of all, if you are alone in lane, unequivocally, you should be pushing. That is to say, if you KNOW the enemy has gone mid or top, and you can safely push, you should be pushing, or joining your team in the teamfight. You are effectively denying the enemy xp when you crash your minions against the enemy turret.

However, if you ARE vs. another enemy, it's more interesting, so I'll speak to that. In that case, it's pretty situation dependent, so I'll give you just a few examples and the philosophy behind them.

All of these assume early game. Late game you should be grouping more, unless you're Sylvanas or something.

Spider Queen:

You are Sonya vs Nazeebo and you have been crushing him.

If Naz is low and the wave is not pushed or pushing, then just sit at the edge of your wave w/in Q range and deny him from XP/Coins while keeping an eye on Midlane to see if they're going to roam and gank you. As SOON as you suspect he's backed, push the wave as hard as you can. Note that the wave must be frozen for this to work, i.e. it is not pushing one direction or the other. In some situations you can bait the enemy to push the wave a little bit, then freeze just out of your tower range so they have to extend into gank range to get exp.

Cursed Hollow:

Valla versus KT

Not much point in denying via freeze on this map (imho) because the map is so large that if someone leaves lane they won't be back in time to reach the minions crashing against the tower.

My personal opinion is that this is a duelist lane, you'll be able to see people leave midlane WAY before they come to gank you top or bot so you don't have to play safe the whole time. I would focus on pushing as hard as you can and trying to outplay the Kael'thas. If you are getting this wave pushed in against 1v1 you are not winning. Of course, if the enemy is MIA from lane (and you suspect they are mid or something) push like mad. You want to crash your minions into their turrets.

Battlefield of Eternity

Anyone vs. anyone

Just don't die and try to soak all the exp, objective is way more important than lanes on this map.

Dragon Shire:

Sylvanas vs. Raynor

Assuming you are mid lane, since Raynor is (arguably) way stronger late game, your mission should be to outpush him pretty hard while landing as many Daggers on him as possible and simultaneously not dying to the many ganks that will come your way.

If you are getting pushed in by Raynor you're having a bad time.

Sky Temple:

Midlane Anubarak vs Valla

The lanes are pretty short on this map, so it's not the end of the world if you're getting pushed in. It's better if you can push the wave to their tower and GTFO, since it's so easy to get ganked on this map. The reality of the situation is, though, the most important thing is XP is even come the time of contesting the lasers, because 3 lasers to none often times leads to a huge XP deficit mid game. With that in mind, the emphasis is on a safe, quick shove, soaking all possible xp, and dodging ganks.

Anyways, that's my piece. Here's the reasoning behind all this stuff: HOTS is predicated around accomplishing as much as possible in a given timeframe, even more so than other MOBAs since you can't get an individual lead, so your goal then is to net as much global xp (and/or things that will LEAD to global xp) in as short a time frame as possible.

Obviously the things that give global XP are minions, towers/keeps, kills, and then things that lead to that are just mercs and objectives (most of the time). By denying you are ALSO creating global xp by creating a deficit for the enemy, so you just have to think in terms of how can I do the most with the amount of time I have.

Often that involves trying to outpush, some of the time that means trying to outduel, some of the time that beings just playing safe, and some of the time that means denying, and the percentages vary by map and team composition. Generally just practice will help you figure out when to do what. MOBAs aren't particularly intuitive to most people, if you know what I mean, so I wouldn't beat yourself up if it takes awhile to get this.

soak can imply A,B and C you just need to be near close enough when the minions die, can be on a bush, can be down the towers, can be when you try to push, you just need to know when you are risking yourself.

there are some heroes that soak in the bush like abathur (sometimes), vikings or nova (when you need to soak).

You've got the basics down, so I just want to say not to overthink things. In the laning phase (pre-10) you just want to be in lane soaking and not endangering yourself. Even if you are melee, you can stand off behind the minions and let them fight other creep waves and just get the xp, because of how the Heroes engine works with shared team xp. Don't cross the "center line" by yourself, meaning halfway between your towers and theirs. That's just asking to get picked or ganked. When an objective is finished, head back to a lane and continue soaking. Particularly if you're solo-queuing, your team's lack of communication will make rotating difficult, so even if you're in a lane 1v1 against another hero who is a bad matchup for yours, fight defensively with your minions back to your towers. They won't be able to push down and you can resume laning when they have to fall back.

alone in a lane just prepare for obj and survive

depends on your character and what the enemy team has and is doing. if they have a gank comp kill the wave and hide so not get caught. if they don't and aren't rotating widdle down the towers. if there is a nova push lane then back off. depends on the map too. did you take SM or CP? then rotate lanes to build up those talents.

Short answer? Push it to the limit. Walk along the razor's edge.

Long answer? Push with your minions as far as is safe. What is safe? Look at the minimap. Can you see all the enemies. If no, can you deduce, based on their last recent position, where they are? If no, then every unaccounted for enemy is about to jump out of the jungle to gank you. Look around. Where is the worst place they can come from? When they do, can you reasonably expect to escape back to safety? If the answer to this is no, you are overextended. Back up until the answer is yes, staying as close to your minion wave as possible to soak or actively clear. This also works for Nova and Zeratul, since they are almost always 'unaccounted for'. Side note: standing in you minion wave gives you time to spot Zeratul's shimmer and cover from Nova snipes, but again "could you make it to safety if..." et cetera, et cetera.

Now the hard part: if you die (ever) then you were someplace you should not have been. If you die, ask yourself how you should have known not to be there, and try to look for those reasons next time you're in lane. There is never 'nothing I could have done', though it is useful to ask the related question 'was it worth it?' before you're too hard on yourself.

In particular, when laning against a single Viking, soak passively so the enemy minions die close to your towers. That way, you can easily harass away the single Viking and deny them the free soak they'd otherwise be getting. Also check for them in the nearby bushes.