There are 51 Trophies that can be collected.

Bronze

Bail Out    
Ejected from a damaged aircraft.     

Guns Guns Guns    
Destroyed 5 hostile vehicles from the air as the door gunner.     

Fire Hazard    
Removed enemy presence from the oil field and the base.     

Life Saver    
Rescued Major Illich from enemy captivity.     

Fearsome Guardian    
Protected your allies using the gunship, and led a successful attack on the base.     

Emergency    
Attempted an emergency landing in a damaged aircraft.     

Total Annihilation    
Disintegrated enemy bombers in mid-air, keeping damage to the city to a minimum.     

Steel Hunter    
Eliminated enemy threat from the Suez Canal.     

Defender of World Heritage    
Prevented enemy from using Trinity to destroy Derbent.     

Hot Pit    
Successfully took off to protect allies amid heavy hostile fire.     

One Million Tons of Scrap Metal    
Sank the enemy fleet and saved the Russian Prime Minister.     

Nick of Time    
Destroyed the ICBMs.     

Berserker    
Destroyed the air defense system, opening the way for allied pilots.

Successive Kill    
Took down 3 hostiles in a row with an assault chain.     

Hard Strike    
Wiped out hostile ground forces with ASM.     

Eagle Eye    
Successfully focused on 3 targets.     

Stay On Target    
Obtained and destroyed 10 enemies in a helicopter.     

Diverse Strikes    
Defeated hostiles with all 3 of the gunship's weapons.     

Gun Master    
Took out a large number of hostiles as the door gunner.   

Shot Master    
Took down 5 hostiles with a direct shot.     

Pursuit Master    
Successfully pursued 3 enemies with counter maneuvers.

Checking In    
Flew through the big hotel's gates.     

Limbo    
Flew beneath a falling chimney.     

Speed Demon    
Destroyed all radar missile defense systems in a short period of time.     

Machine Gun Faithful    
Destroyed an enemy aircraft using only the standard machine gun while piloting an aircraft.

Bomber Master    
Defeated a large number of enemies while flying a bomber.

Nice Assist    
Used DFM support or ASM support to join a flight member.     

Nice Save    
Saved an ally being targeted by hostile players in DFM in competitive online.

The Collector    
Piloted all aircraft in the game.     

Welcome to Mission Co-Op    
Cleared 1 level in mission co-op.     

Aerial Sniper    
Used DFM to destroy 10 airborne enemies in competitive online.     

Ground Pounder    
Used ASM to destroy 30 ground enemies in competitive online.     

Interceptor    
Used an attack helicopter and destroyed 10 missiles or bombs in competitive online.

Silver

Critical Hit    
Took down a hostile fighter with a charged homing missile.

Switch Master    
Successfully switched from defense to offense 10 times with counter maneuvers.

Smooth Flight    
Managed to get Sova 1 to land safely.

Chain Master    
Took down 5 hostiles in a row with an assault chain.     

Nice Kill    
Shot down a large number of hostile players in competitive online.

Veteran Pilot    
Sortied for 50 hours in competitive online or mission co-op.

Formation Attack    
Joined the same DFM with two others and shot down a hostile (mission co-op).

World Tour    
Played on all maps in competitive online.     

Ace of Aces    
Obtained an A rank for every mission (free mission).     

All Rounder    
Obtained all MVPs.

Gold

Warwolf 1    
Cleared all missions.

Warwolf Squadron    
Cleared all levels in mission co-op.

6 Secret

http://www.ps3trophies.org/game/ace-combat-assault-horizon/trophies/