Key
SSJ= Super Saiyan
ASSJ= Ascended Super Saiyan
SSJ2= Super Saiyan level 2
SSJ3= Super Saiyan level 3
SSJ4= Super Saiyan level 4

Goku
SSJ- When he fought Freeza in Planet Namek. He saw 
Krillin die and became enraged.
ASSJ- While training in the room of spirit and time in 
the Cell Saga. SSJ2- In the after life while training.
SSJ3- Also reached this level in the after life
SSJ4- While he was in Golden Oozaro form, he saw Pan 
cry and then turned SSJ4. This happened during the 
Bebi Saga.

Gohan
SSJ- While training in the room of spirit and time. He 
badly wanted to help everyone which triggered him to 
become SSJ.
SSJ2- When he fought Cell in the Cell Game. He saw 
Android 16 die then became enraged.

Vegeta
SSJ- While training on a planet, a meteor shower came 
down towards the his ship and the only way he could 
save the ship and leave the planet was to protect the 
ship. And so then became super saiyan trying to fight 
off all the meteors.
SSJ2- He reached this as he became Majin Vegeta.
SSJ3- He skipped the level.
SSJ4- Bulma used a ray that turned him into his Golden 
Oozaru form then went SSJ4.

Future Trunks
SSJ- After seeing everyone die in his timeline he got 
this level.
ASSJ- In the room of spirit and time during the Cell 
Saga.

Chibi-Trunks
SSJ- Reached this level when he tried it for the first 
time.

Goten
SSJ- Reached this level when he tried it for the first 
time.