There are 12 Trophies that can be collected.

BRONZE

Falling Stone
Crush 20 workers with one giant boulder. (Campaign, Survival)

Great balls of fire
Damage 5 tower blocks with one meteor shower. (Campaign, Survival)

Holy wave
Use a tidal wave that spares cursed jar bearers. (Campaign, Survival)

Blown away
Use a sand storm to send a worker flying far away. (Campaign, Survival)

Full house
Kill 10000 workers over multiple sessions. (Campaign, Survival)

Deity
Kill 50 workers. (Campaign, Survival)

Break the ice
Kill 30 workers frozen by the other player in a coop game.

Cursed!
Destroy 15 cursed jars in a single game. (Campaign, Survival)

SILVER

Babel Falling
Complete the campaign.

Cruel god
Send 10 maximum difficulty waves to your opponent in a versus multiplayer match.

Apocalypse
Earn the highest distinction by performing power combos. (Campaign, Survival)

GOLD

Survivor
Survive 5 minutes when the tower is almost complete. (Campaign, Survival)

http://www.ps3trophies.org/game/babel-rising/trophies/