There are 12 Trophies that can be collected.

Bronze

The Law is Dead    
Kill your first Damned Zombie.     

Alone at the End    
Complete Story Mode in Single Player.     

Hard Worker    
Complete all Objectives listed for any chapter.     

Got Pwnage?    
Use each character's Pwnage ability 10 times.     

Weapons Expert    
Pick-up all five mash-up weapons with each character.     

Money Grubber    
Collect $20,000 in a single chapter.     

Beacon in the Dark    
Survive a fortnight in Blackout Mode.

Silver

Long Term Survival Plan    
You survived 30 Days in Survival: Classic.     

Super Savior    
Bring all possible survivors to Def Money's yacht in Story Mode.     

Playing with Power    
Get all 100 upgrades for your characters.     

Epic Run    
Complete a Story Mode: New Game with 3 other players.

Gold

Exhausted    
Complete All Chapter Objectives.

http://www.ps3trophies.org/game/zombie-apocalypse-never-die-alone/trophies/