There are 32 Trophies that can be collected.

Bronze

Valley Wanderer    
Defeat 1st Colossus    

The Sloth    
Defeat 2nd Colossus    

Disturbed Sleep    
Defeat 3rd Colossus    

Path of Gravestones    
Defeat 4th Colossus    

Aerial Dance    
Defeat 5th Colossus    

Entombed Giant    
Defeat 6th Colossus    

Waves of Thunder    
Defeat 7th Colossus    

Wall Scaler    
Defeat 8th Colossus    

Slumbering Caveman    
Defeat 9th Colossus    

Unknown Tracks    
Defeat 10th Colossus    

Guardian Unleashed    
Defeat 11th Colossus    

Silent Thunder    
Defeat 12th Colossus    

Signs in the Sky    
Defeat 13th Colossus    

Shielded Colossus    
Defeat 14th Colossus    

Valley of the Fallen    
Defeat 15th Colossus    

Final Colossus    
Defeat 16th Colossus

Hang Glider    
Hang onto the hawk for more than 30 seconds

Slippery Ride    
Hang onto the fish for over 30 seconds

Silver

The Forbidden    
Clear the game in normal difficulty    

Wander and the Forbidden    
Clear the game in hard difficulty

Resistance    
Endure being sucked into the light for over 1 minute in the ending    

Collector    
Acquire all items available in normal time attack

Agro Circus    
Perform all stunt riding using Agro

Tower of Prayer    
Save at all save points

Gold

Climber    
Reached the top of the shrine

Meticulous Collector    
Acquire all items available in hard time attack

Cornucopia    
Acquired all fruits in the world

Endangered Lizards    
Killed all shining lizards    

Stalwart Wander    
Max out HP bar    

Mighty Wander    
Max out stamina bar

Platinum

Wander and the Colossus    
Acquired all Trophies

http://www.ps3trophies.org/game/team-ico-collection-shadow-of-the-colossus/trophies/