There are 15 Trophies that can be collected.

Bronze

ESCAPE    
Clear CHAPTER 1 on any difficulty.    

LOST    
Clear CHAPTER 2 on any difficulty.    

EMPTINESS    
Clear CHAPTER 3 on any difficulty.    

DESPAIR    
Clear CHAPTER 4 on any difficulty.    

REUNION    
Clear CHAPTER 5 on any difficulty.    

HOPE    
Clear CHAPTER 6 on any difficulty.    

EAT THIS!!!    
During CHAPTER 1 Boss encounter, throw a grenade and damage JUSTICE.    

CUT THE WIRE    
During CHAPTER 2, cut the wire and let the enemies fall.    

BONUS EXPLORER    
Find a secret room.    

GET THE BUS    
Shoot and destroy the BUS.

WHO'S NEXT?    
See the bad ending of HOD4 SP.    

FIGHT BEGINS    
See the good ending of HOD4 SP.    

ANOTHER PANDORA'S BOX    
See the true ending of HOD4 SP.

Silver

TOP AGENT    
Get RANK 2 or above in Final Result.

Gold

PANDORA'S BOX    
See all 4 endings of THE HOUSE OF THE DEAD 4.

http://www.ps3trophies.org/game/the-house-of-the-head-4/trophies/