The common ones besides death saves are hindered attacks. When grappled you have to make a DC5 flat check in order to get an attack in (grappled being different from restricted, which means you’ve essentially been pinned and are unable to attack).

Attacking a hidden creature (you know roughly where it is) requires you make a DC 11 flat check.

Attacking an undetected creature (no idea where it is at all) requires you make the same flat check but rolled blindly, so you won’t know if your damage actually hits the creature or not.

There are other cases for flat checks, some random encounter tables use them for example, but they mostly tend to be GM oriented rather than player oriented.

Yeah, in PF2E a flat check is used when you want something to be completely unmodified. A DC10 flat check would be a 55% chance for everyone.