There are 15 Trophies that can be collected.

Bronze

Space adventurer
Finishing the outside of the Nahirg’s planet level.

As if you owned the place
Completing the Nahirg's base level.

Spider killer
Finishing the cave level.

This is getting hot
Finishing the rock and lava cave level.

Treasure finder
Find 5 nahirg treasures.

Frisbee player
Killing 100 enemies with the blade launcher.

Ice cream man
Use frozen air to freeze 50 enemies.

Toaster
Kill 100 enemies using the flame thrower.

Avalanche!
Use rock spikes to kill 50 enemies.

Rocky
Use the power of your rock punch to kill 50 enemies.

Variety is the spice of life
Kill 50 enemies using each of Zack’s powers.

You can’t scare me
Killing a Nahirg general using only the blade launcher.

Nothing can stop me
Finish the game in any difficulty level.

Silver

I feel the power
Reach Zack's suit level 20.

Gold

Archaeologist
Finding all Nahirg treasures.

http://www.ps3trophies.com/forums/playstation-trophy-news/9443-zack-zero-trophy-list.html