As has already been alluded to:
  1. Anything that makes you obviously monstrous (look like a troll, Marvel's The Thing, etc.) would have serious social complications.
  2. Anything that makes you innately hazardous (anything from being on fire to being able to cast magic missile) would have social and emotional/mental well-being consequences (how many people who have ever had suicidal ideation would have real problems with a lethal power which could not be taken away?).
  3. Anything that gives you massive control over others, or the world in general, could have self-identity concerns (I don't love that people die from preventable reasons, but in general I don't have a lot of control over that fact. If I did...).
  4. Anything that allows you to know things you'd rather not know (knowing what people really thought, knowing when and how you and your loved ones will die), or perhaps not be able to turn off (being a telepath in a large city)

This is all part and parcel of the superhero genre (teenagers discover that they are mutants, The Thing, 'with great power comes great responsibility,' etc.) as well as classical mythology (examples being Herakles, Cassandra, etc.), and I think it has been explored exhaustively there.