There are 51 Trophies that can be collected.

Bronze

You Will Be Ace    
Destroy 4 planes in Versus mode.

Rookie    
Earn the second rank.    

Combat Pilot    
Earn the fifth rank.

Preparing    
Complete the Pre-War chapter.

Pearl Harbor    
Complete the Japanese chapter "Attack on Pearl Harbor".    

Wake Island    
Complete the American chapter "Battle of Wake Island".    

Midway Atoll    
Complete the Japanese chapter "Battle of Midway".    

Coral Sea    
Complete the American chapter "Battle of the Coral Sea".    

Guadalcanal    
Complete the Japanese chapter "Guadalcanal Campaign".    

Winner    
Complete any Dynamic campaign.    

Malta    
Complete any mission in the Single Missions chapter "Siege of Malta".    

Kuban    
Complete any mission in the Single Missions chapter "Battle for the Kuban".    

Ruhr    
Complete any mission in the Single Missions chapter "Battle of Ruhr".    

New Guinea    
Complete any mission in the Single Missions chapter "New Guinea Campaign".    

It Was Easy    
Land on an aircraft carrier.

Choose Your Favourite   
Fly in one of each country's planes.    

Make a Flight    
Complete any mission in CO-OP mode.    

New Skin    
Earn any skin for your plane.    

New Decal    
Earn any decal for your plane.

First Wave    
Destroy all vehicles on Wheeler Field in the mission "First Wave".    

Second Wave    
Bomb two different targets by one loadout in the mission "Second Wave".    

Return to Oahu    
Make "Touch and Go" on Ford Island in the mission "Return to Oahu".    

Survivor    
Land your damaged aircraft on the airfield in the mission "The Two Against Thirty-Nine".    

Second Breath    
Rearm on the airfield between attacks on hostile ships in the mission "American Counterattack".    

Blitz Attack    
Destroy 6 enemy bombers before they reach the Wake in the mission "Japanese Raid".    

Bombshell    
Damage destroyer with machine gun fire in the mission "Invasion of Tulagi".    

Sentinel    
Don't let enemy fighters destroy any bomber of your group in the mission "Attack on Lexington".    

Untouchable    
Don't die in the mission "Shokaku Defense".    

Not a Victim    
Destroy two hostile fighters with the rear gunner in the mission "To Scratch One Flat-top".    

Eagle-Eyed    
Find the TF-17 carrier fleet in the "Counterattack" mission.    

Hedgehopper    
Destroy 5 light targets on Midway Island on Simulator difficulty in the "Midway Atoll" mission.    

Defender    
Destroy 3 enemy bombers in each wave on Realistic difficulty in the "First Carrier Fleet" mission.    

Way of the Sun    
Complete the 1st secondary mission objective in the "Fate of Hiryu" mission.    

Formation Keeper    
Stay near the flight leader all the way to the enemy fleet in the "American Strike" mission.    

One of a Few    
Land on the Zuikaku at the end of the "Battle of South Pacific" mission on Realistic difficulty.    

Fast and Furious    
Complete the first objective in less than 5 minutes in the "Tulagi Landing" mission.    

Total Annihilation    
Destroy at least 20 enemy bombers in the "Battle of the Santa Cruz" mission.    

Sharp Shooter    
Destroy 5 enemy planes on Realistic difficulty in the "Battle of the Eastern Solomons" mission.    

Killer    
Kill 15 enemy infantry units in the "Battle for Henderson Field" mission.

Silver

Good Start    
Receive 5 medals.    

Brave Spirit    
Win a battle against superior forces in Versus mode.    

Leader    
Win a battle in Versus mode as the leader.

Half-Way    
Earn the tenth rank.

The Italian    
Fly in each Italian plane.    

The Australian    
Fly in each Australian plane.

New Record    
Destroy 100 planes.    

New Weapon    
Destroy 100 ground units.

Gold

To The Far East    
Complete the American Historical campaign.    

From The Far East    
Complete the Japanese Historical campaign.

Platinum

Bird of Steel    
Earn all available Trophies for Birds of Steel.

1 Secret

Bronze

Take-Off    
Take off.

http://www.ps3trophies.org/game/birds-of-steel/trophies/