From now on, until I get a good argument otherwise, I'm gonna go ahead and say purity has nothing to do with it. It's simplt: 1 - You have to have high enough power, and 2 - You need some angering event to set you off into a rage.
Chronologically...
Future Gohan: The deaths of his friends, particularly Piccolo and Kuririn (Or the thoughts of their deaths once he attained anough power... I can't remember)
Future Trunks: The death of Future Gohan (Or, in the manga, simply anger at the insidiousness of 17 and 18)
Son Goku: Piccolo's near death and Kuririn's second death (He was unaware of Polunga's ability to revive people more than once remember)
Vegeta: The idea of always being number 2 at best made his anger explode. Remember my inferiority complex theory?

Son Gohan: The idea of not being able to protect his friends, and his failure to do so in the past.
Goten and Present Trunks: ... uh... Magic.
I won't bother trying to analyze why the SS2 and 3 tranformations happened... I imagine it's the same anger + power combo. The only clear cut one is Gohan, with 16's great speech and subsequent death, plus the Cell Juniors beating up his comrades.
That's all. Go Red Sox!