So this has kinda been bugging me for a while: underage magic/ unforgivable curses

So we all know in book 2, Dobby does an inflatable spell and HP immediately gets an owl from the ministry for improper use of magic

1: it wasn’t him

2: They checked his wand to see the last spell that was used

If unforgivable cursed a are illegal, and they can trace underage magic, then it shouldn’t be difficult to recognize these curses and trace them down

I know it’s different with magic households, cuz magic only shares the area it’s in. It’s stated in the books. But why doesn’t that stop kids I wizardikg households to use magic anyway? By the books definition, they wouldn’t be caught.

I’m sure it’s up to the parents to enforce the rule but, it doesn’t really make sense to me