There are 14 Trophies that can be collected.

BRONZE

The Onslaught
Finish Act I (single player).

Air Superiority
Shoot down 300 fighters in Campaign mode.

A Private Party
Shoot down all V-1 rockets headed for London (single player).

God Save the King
Fly under every bridge in London (single player).

Divine Thunder
Defend all battleships at Iwo Jima (single player).

Ace of the Skies
Shoot down 6 planes in under a minute.

Moving Target
Drop a bomb on a plane during takeoff.

A Confirmed Kill
Shoot down an enemy fighter with a rocket.

Trigger Happy
Launch more than 100 rockets in a single mission.

Two Heads are Better Than One
Play a co-op mission.

Test Pilot
Finish a single mission in Simulation mode (other than Baptism of fire).

SILVER

War Hero
Earn all medals and the highest rank in all missions (Act I and II).

Road to Victory
Finish Act II (single player).

GOLD

V for Victory
Finish all missions on HARD difficulty level (Act I and II).

http://www.ps3trophies.org/game/dogfight-1942/trophies/