One thing that the ASOIAF is filled with is GRRM's different takes on warrior women. Due to the sheer number, the post became massive to I decided to break it up into 4 sections that I am going to post this week. Part I was about wildings, Part II was about members of Houses (and bastards), Part III will feature dragonriders and Part IV will have all the others (outlaws, sellswords, etc.).

Link to Part I: Wildings

Link to Part II: Highborn Characters

Link to Part IV: Sellswords, Outlaws, Etc.

Warrior Women of Ice and Fire Part III: Dragonriders

As I mentioned in Part I. My initial name for this post was "Warrior Maids" but Chella (featured in Part III) made (no pun intended) me change my mind:

Tyrion licked his fingers. "With pleasure," he said. "The fair maid is Chella daughter of Cheyk of the Black Ears.""I'm no maid," Chella protested. "My sons have taken fifty ears among them." -AGOT, Tyrion VII

Note: Not all of these dragonriders are known to have used these dragons to fight/battle, but I still feel it is necessary to include them all. Even just due to the potential for devastation they can cause at any moment:

A dragon's natural span of days is many times as long as a man's, or so the songs would have us believe . . . but the dragons the Seven Kingdoms knew best were those of House Targaryen. They were bred for war, and in war they died. -ASOS, Daenerys I

Aegon's Invasion

Aegon and his wisters invaded the seven kingdoms with three monstrous dragons:

You could have ridden a horse down Vhaghar's gullet, although you would not have ridden it out again. Meraxes was even bigger. And the greatest of them, Balerion, the Black Dread, could have swallowed an aurochs whole, or even one of the hairy mammoths said to roam the cold wastes beyond the Port of Ibben. -AGOT, Tyrion II

Visenya Targaryen (Vhagar)

While a dragonrider, Visenya was not only a true warrior woman (she carried the valyrian sword Dark Sister), she was also a dabbler in black magic.

Rhaenys Targaryen (Meraxes)

Much different than her sister, Rhaeynys was much less of warrior but she also flew more than her brother/sister.

Both Visenya and Rhaenys were instrumental in the different battles of Aegon's Invasion, primarily the Field of Fire and the Dornish Wars.

The Dance of the Dragons I

Rhaenys Targaryen - Meleys

Rhaenys rode Meleys during the first Dance of the Dragons. Rhaenys and Meleys defended Rook's Rest from Criston Cole, but they were killed when Sunfyre and Vhagar (the largest living dragon showed up):

That aid arrived in the form of Princess Rhaenys—then five-and-fifty, but as fearless and determined as she had been in her youth—and her dragon Meleys, the Red Queen. But Cole had brought dragons too—for Aegon II himself arrived on the field upon Sunfyre, and his brother Aemond One-eye rode Vhagar, the greatest living dragon.

It is recounted that Princess Rhaenys, the Queen Who Never Was, did not shrink from her foe. With a glad cry and a crack of her whip, she sent Meleys flying up to face them. Only Vhagar and Aemond came out of that battle unscathed; Sunfyre was crippled, and King Aegon II barely survived, suffering broken ribs, a broken hip, and burns that covered half his body. Worst was his left arm, where the dragonfire melted the king's armor into his flesh. Rhaenys's body was found several days later amidst the ruin of the Red Queen's corpse, but so burned as to be unrecognizable. -TWOIAF, The Targaryen Kings: Aegon II

Baela Targaryen - Moondancer

Baela refused to leave Dragonstone during the Dance of the Dragons, wishing to fight alongside her betrothed Jacaerys. When Dragonstone fell to Aegon II, Baela mounted Moondancer and met Aegon and Sunfyre in the sky:

King Aegon found many around Dragonstone who had grievances against Rhaenyra—for the loss of sons, husbands, and brothers in her war, or for slights they imagined—and with their aid he conquered Dragonstone. It took no more than an hour, largely unopposed as it was...except for Prince Daemon's daughter, the fourteenyear-old Baela Targaryen and her young dragon, Moondancer. Baela had escaped the men who tried to seize her and had made her way to her dragon. And as Aegon II sought to land in the courtyard of the castle on Sunfyre, thinking himself triumphant, the dragon and the princess rose to meet him.

Moondancer was much smaller than Sunfyre, but also much swifter and far more nimble, and neither the dragon nor the princess on her back lacked courage. The dragon swooped and clawed and snapped at Sunfyre, raking and tearing until at last a blast of flame blinded the beast. Tangled together, the two dragons fell, and their riders with them. Aegon II leapt at the last moment from Sunfyre's back, both legs shattering, while Baela remained with Moondancer to the bitter end. When Alfred Broome drew his sword to kill her where she lay broken and unconscious, Ser Marston Waters tore the sword from his grasp and carried her to the maester, saving her life. -TWOIAF, The Targaryen Kings: Aegon II

Rhaenyra Targaryen - Syrax

The Half-Year Queen/Realm's Delight, she rode Syrax while was warring against her younger half brother Aegon II in the Dance of the Dragons:

Meanwhile, Prince Daemon Targaryen himself hastened south on the wings of his dragon, Caraxes. Flying above the western shore of the Gods Eye, well away from Ser Criston’s line of march, he evaded the enemy host, crossed the Blackwater, then turned east, following the river downstream to King’s Landing. And on Dragonstone, Rhaenyra Targaryen donned a suit of gleaming black scale, mounted Syrax, and took flight as a rainstorm lashed the waters of Blackwater Bay. High above the city the queen and her prince consort came together, circling over Aegon’s High Hill.

The sight of them incited terror in the streets the city below, for the smallfolk were not slow to realize that the attack they had dreaded was at last at hand. Prince Aemond and Ser Criston had denuded King’s Landing of defenders when they set forth to retake Harrenhal … and the Kinslayer had taken Vhagar, that fearsome beast, leaving only Dreamfyre and a handful of half-grown hatchlings to oppose the queen’s dragons. The young dragons had never been ridden, and Dreamfyre’s rider, Queen Helaena, was a broken woman; the city had as well been dragonless. -The Princess and the Queen

Was later fed to Sunfyre:

Aegon II, ever spiteful of his sister and enraged at the agony of his shattered legs and the impending death of his dragon, fed Rhaeynra to Sunfyre before the eyes of her sole surviving son (so far as any man or woman in the Seven Kingdoms knew), Aegon the Younger. So passed the Realm's Delight, the Half-Year Queen, on the twenty-second day of the tenth moon of 130 AC. -TWOIAF, The Targaryen Kings: Aegon II

Nettles - Sheepstealer

Nettles was a dragonseed (debatable) who used mutton (and/or sorcery) to bond with the wild dragon Sheepstealer.

Nettles fought bravily on Queen Rhaenyra's behalf during the Battle of the Gullet and was present for the fall of King's Landing. Later was declared a traitor and fled to the Vale where they killed 16 men and injured 60 more before fleeing deeper into the Mountain's of the Moon.

The last thing we know about her is that she likely was a "fire-witch" worshipped by the Burned Men:

Most of these clan names have some meaning, however obscure those meanings might be to us. The Black Ears take the ears of men they defeat in battle as trophies, we know. Amongst the Burned Men, a youth must give some part of his body to the fire to prove his courage before he can be deemed a man. This practice might have originated in the years after the Dance of the Dragons, some maesters believe, when an offshoot clan of the Painted Dogs were said to have worshipped a fire-witch in the mountains, sending their boys to bring her gifts and risk the flames of the dragon she commanded to prove their manhood. -TWOIAF -The Vale

Other Battles

Aerea Targaryen - Balerion

As far as I know Aerea didn't fight in any wars, but when she fled on Balerion, its likely that was one of the few people to see the Doom of Valyria where they were attacked by (likely) firewyrm(s):

"Firewyrms. Some say they are akin to dragons, for wyrms breathe fire too. Instead of soaring through the sky, they bore through stone and soil. If the old tales can be believed, there were wyrms amongst the Fourteen Flames even before the dragons came. The young ones are no larger than that skinny arm of yours, but they can grow to monstrous size and have no love for men." -AFFC, Arya II

Some of Septon Barth's thoughts:

In Septon Barth's Dragons, Wyrms, and Wyverns, he speculated that the bloodmages of Valyria used wyvern stock to create dragons. Though the bloodmages were alleged to have experimented mightily with their unnatural arts, this claim is considered far-fetched by most maesters, among them Maester Vanyon's Against the Unnatural contains certain proofs of dragons having existed in Westeros even in the earliest of days, before Valyria rose to be a power. -TWOIAF, Beyond the Free Cities: Sothoryos

and:

It has been three days since the princess perished, and I have not slept. I do not know that I shall ever sleep again. The Mother is merciful, I have always believed, and the Father Above judges each man justly... but there was no mercy and no justice in what befell our poor princess. How could the gods be so blind or so uncaring as to permit such horror? Or is it possible that there are other deities in this universe, monstrous evil gods such as the priests of Red R'hllor preach against, against whose malice the kings of men and the gods of men are naught but flies? I do not know. I do not want to know. If this makes me a faithless septon, so be it. - Fire & Blood, Jaehaerys and Alysanne - Their Triumphs and Tragedies.

Non-Combatant Riders

Im sure that numerous of these characters did experience some form of combat, if you notice any that I missed, please let me know!

Jaenara Belaerys - Terrax

A noblewoman of Valyria, she used her dragon to explore:

We do not even know the true size of Sothoryos. Qartheen maps once showed it as an island, twice the size of Great Moraq, but their trading ships, venturing farther and farther down the eastern coats, were never able to find the bottom of it. The Ghiscari who settled Zamettar and Gorosh believed Sothoryos to be as large as Westeros. Jaenara Belaerys flew her dragon, Terrax, farther south than any man or woman had ever gone before, seeking the boiling seas and steaming rivers of legend, but found only endless jungle, deserts, and mountains. She returned to the Freehold after three years to declare that Sothoryos was as large as Essos, "a land without end." -TWOIAF, Beyond the Free Cities: Sothoryos

Since she was gone for three years, its possible she encountered/fought one of the following that is known to exist in Sothoryos:

Nor is disease the only danger that those who seek to know this wet, green land must face. Huge crocodiles lurk beneath the surface of the Zamoyos and have been known to overturn boats, swimming up from below so they might devour their occupants as they struggle in the water. Other streams are infested by swarms of carnivorous fish capable of stripping the flesh from a man's bones in minutes. There are stinging flies, venomous snakes, wasps and worms that lay their eggs beneath the skins of horses, hogs, and men alike. Basilisks both great and small are found in great numbers on Basilisk Point, some twice the size of lions. In the forests south of Yeen, there are said to be apes that dwarf the largest giants, so powerful they can slay elephants with a single blow.

Farther south lie the regions known as the Green Hell, where beasts even more fearsome are said to dwell. There, if the tales are to be trusted, are caverns full of pale white vampire bats who can drain the blood from a man in minutes. Tattooed lizards stalk the jungles, running down their prey and ripping them apart with the long curved claws on their powerful hind legs. Snakes fifty feet long slither through the underbrush, and spotted spiders weave their webs amongst the great trees.

Most terrible of all are the wyverns, those tyrants of the southern skies, with their great leathery wings, cruel beaks, and insatiable hunger. Close kin to dragons, wyverns cannot breathe fire, but they exceed their cousins in ferocity and are a match for them in all other respects save size.

Brindled wyverns, with their distinctive jadeand-white scales, grow up to thirty feet long. Swamp wyverns have been known to attain even greater size, though they are sluggish by nature and seldom fly far from their lairs. Brownbellies, no larger than monkeys, are even more dangerous than their larger kin, for they hunt in packs of a hundred or more. But most dreaded of all is the shadow-wing, a nocturnal monster whose black scales and wings make him all but invisible...until he descends out of the darkness to tear apart his prey.

Alysanne Targaryen - Silverwing

While there are stories of her battling beyond the wall, I believe that it was confirmed that this was just a story for the children:

On the third day of the third moon of 129 AC, while entertaining Jaehaerys and Jaehaera from his bed with a tale of their great-great-grandsire and his queen battling giants, mammoths, and wildlings beyond the Wall, the king grew tired. He sent his grandchildren away when the tale was done and fell into a sleep from which he never awoke. He had ruled for six-and-twenty years, reigning over the most prosperous era in the history of the Seven Kingdoms but seeding within it the disastrous decline of his house and the death of the last of the dragons. -TWOIAF, The Targaryen Kings: Viserys I

Alyssa Targaryen -Meleys

Alyssa claimed Meleys in 75AC, and we know she took both of her sons riding (Viserys I in 77AC and Daemon in 81AC). But she died birthing in 84 AC.

Laena Velaryon - Vhagar

Laena became a dragonrider in 105 AC, claiming the largest living dragon. Laena loved to fly and possibly could have experienced some form of combat on her travels across the Narrow Sea, but she died in 120 AC several years before the Dance of the Dragons.

Jaehaera Targaryen - Morghul

While bonded, Jaehaera never rode Morghul (she was not big enough yet). Morghul was slain during the storming of the Dragonpit (likely by the Burned Knight). Morghul killed scores of people before dying.

Rhaena Targaryen - Dreamfyre

Bonded in 35AC when Rhaena was twelve. Was forbidden to ride Dreamfyre on a royal progress since her brusband had not yet claimed a dragon.

When her husband finally claimed a dragon in 43AC and went to war against their uncle Maegor, he was killed but Dreamfyre/Rhaena were no present (likely wouldn't have mattered vs. The Black Dread).

Tried to hide in various places from Maegor, taking the dragon to Fair Castle, Casterly Rock, etc. Also used Dreamfyre to look for the missing Aerea Targaryen (her daughter).

Helaena Targaryen - Dreamfyre

During Viserys I reign, Helaena mounted Dreamfyre by the age of eleven. When her 6 year old son Jaehaerys was killed in the war, she spent her days in darkness, weeping and was not capable of helping the war effort/riding Dreamfyre.

Dreamfyre was later killed in the storming of the dragonpit, but killed more men the other three dragons combined.

Rhaena Targaryen - Morning

In 135 AC a 5 or 6 year old Morning was finally big enough for Rhaena to ride. Morning was one of four dragons to survive the Dance of the Dragons. We will likely find out more about Morning/Rhaena in Fire & Blood II, but she (Morghul) likely died/disappeared before 153 AC.

The Future

Daenerys Targaryen - Drogon

Not much need for an explanation, Daenerys has bonded with Drogon and is about to go all "Fire and Blood".

Other

Personally, I expect the other dragonriders in the series to be male, that said, the possibility does allow for a female character with valyrian blood(or some possible alternative methods) to ride a dragon.

TLDR: List of Warrior Women (Part III): Dragonriders.