There are 50 Trophies that can be collected.

Bronze

Destiny Begins    
Select a Power.     

Think About It…    
Make first power destiny choice.     

Choose Wisely    
Make second power destiny choice.     

The Choice Is Made    
Make third power destiny choice.

Splicer    
Equip your first X-Gene.     

At Least It's Aerodynamic…    
Equip your first suit.     

I've Got the Power    
Fully level up a power.

Four of a Kind    
Equip a complete X-Gene set and suit.     

Beginner's Luck    
Complete your first Great combo.     

This can't be happening!    
Complete your first Insane combo.     

How Strong Could It Be?    
Trigger X-Mode.     

Broken Glass, Everywhere…    
Break 30 Combat Text Pop Ups.     

Fight Terror with Terror    
Defeat 10 enemies with one Ultra power.     

Side-tracked    
Complete your first challenge mission.     

Taking Every Opportunity    
Complete 10 unique challenge missions.

Garbage Collection    
Destroy your first piece of propaganda.

Profiler    
Collect your first dossier.

Purify the Purifiers    
Defeat 2000 Purifiers.     

Mechageddon    
Defeat 20 Purifier Stalker Mechs.     

Reinforced    
Defeat 20 Prime Enforcers.     

U Mad, Bro?    
Defeat 30 U-Men in the Secret Lab.     

The Goon Squad    
Defeat 500 MRD Troops.

Silver

Got My Eye on You    
Join Cyclops and the X-Men.     

Magneto Is Right    
Join Magneto and the Brotherhood.

Fully Evolved    
Fully level up all powers.

Completionist    
Complete all 15 unique challenge missions.

Cleaned up the City    
Destroy 25 pieces of propaganda.

Archivist    
Collect 15 dossiers.

Gold

Beta Level Mutant    
Finish the game on New Mutant difficulty.     

Alpha Level Mutant    
Finish the game on X-Man difficulty.     

Omega Level Mutant    
Finish the game on X-treme difficulty.

Platinum

Homo Superior    
Collect all other trophies.

18 Secret

http://www.ps3trophies.org/game/x-men-destiny/trophies/