There are 37 Trophies that can be collected.

BRONZE

Wings Earned
Complete a Mission in Campaign mode (Singleplayer or Co-Op).

Bombardier
Sucessfully bomb a target in Campaign mode (Singleplayer or Co-Op).

Fish In The Water
Sucessfully torpedo a target in Campaign mode (Singleplayer or Co-Op).

Daredevil
Sucessfully divebomb a target in Campaign mode (Singleplayer or Co-Op).

Killing Spree
Destroy 50 enemies in Campaign mode (Singleplayer or Co-Op).

No Mercy
Destroy 250 enemies in Campaign mode (Singleplayer or Co-Op).

Hellcat Pilot
Unlock the Hellcat in Campaign mode (Singleplayer or Co-Op).

Day of Infamy
Complete the Day of Infamy mission in Campaign mode (Singleplayer or Co-Op).

Battle of Midway
Complete the Battle of Midway mission in Campaign mode (Singleplayer or Co-Op).

A Million Men
Complete the Bloody Tarawa mission in Campaign mode(Singleplayer or Co-Op).

End of the Kido Butai
Complete the End of the Kido Butai mission in Campaign mode (Singleplayer or Co-Op).

Rookie
Complete Campaign mode on Easy Difficulty or higher (Singleplayer or Co-Op).

Be My Wingman
Destroy an enemy aircraft that is targetting a team mate in Co-Op Campaign mode.

First Blood
Destroy an enemy in any match in Multiplayer mode.

Survivor
Win an Elimination match in Multiplayer mode.

Dogfighter
Win a Dogfight match in Multiplayer mode.

Team Player
Win a Team Dogfight match in Multiplayer mode.

Sinker
Win a Scratch One Flattop match in Multiplayer mode.

SILVER

Fish In A Barrel
Sink 25 ships (Singleplayer or Co-Op).

Relentless
Destroy 500 enemies (Singleplayer or Co-Op).

Master of Arms
Destroy an enemy plane with rockets (Singleplayer or Co-Op).

Veteran
Complete Campaign mode on Normal Difficulty or higher (Singleplayer or Co-Op).

Wingman
Complete a Co-Op Campaign mission as a client.

Wing Leader
Complete a Co-Op Campaign mission as a host with one or more team mates.

Untouchable
Destroy 5 enemies without dying in a single match in Multiplayer mode.

Last Man Standing
Win 5 Elimination matches in Multiplayer mode.

Alpha Dog
Win 5 Dogfight matches in Multiplayer mode.

Pack Leader
Win 5 Team Dogfight matches in Multiplayer mode.

Homewrecker
Win 5 Scratch One Flattop matches in Multiplayer mode.

GOLD

Superbattleship
Sink the Musashi in Campaign mode (Singleplayer or Co-Op).

Ace
Complete Campaign mode on Hard Difficulty (Singleplayer or Co-Op). Prestige

Immortal
Destroy 10 enemies without dying in a single match in Multiplayer mode.

PLATINUM

Damage Inc. Pacific Squadron WWII
Collect all other trophies in the game.

4 SECRET

http://www.ps3trophies.org/game/damage-inc-pacific-squadron-wwii/trophies/