Here is the full list of all 34 Battleship trophies - there are 15 bronze, 14 silver, 4 gold and 1 platinum.
Collect all Trophies.
Complete "The Arrival" on any Difficulty.
Complete "Construction" on any Difficulty.
Complete "Overrun" on any Difficulty.
Complete "The Big Guns" on any Difficulty.
Complete "They're Back!" on any Difficulty.
Complete "Retaliation" on any Difficulty.
Complete "End Game" on any Difficulty.
Complete the game on any Difficulty.
Complete the game on Captain Difficulty or higher.
Complete the game on Admiral Difficulty.
Used your first support call.
Occupied all support positions in "Overrun".
Downed a Thug with the pistol.
Downed 3 or more enemies with one shot from the Railgun.
Downed 5 enemies with one support call.
Disarm or plant 20 charges.
Defeated 40 enemy vessels in Ship Control gameplay.
Fully outfit a ship with the maximum number of Wild cards available.
Down 40 enemies that are stunned by the LRAD.
Down 40 enemies with the Pistol.
Down 40 enemies with the KRAW.
Down 40 enemies with the Carbine.
Down 40 enemies with the Shotgun.
Down 40 enemies with the Alien Railgun.
Down 40 enemies with Grenades.
Destroy 10 enemy ships with the USS Chesapeake.
Destroy 20 enemy ships with the USS John Quincy Adams.
Destroy 20 enemy ships with the USS Laredo.
Destroy 20 enemy ships with the USS Yukon.
Destroy 20 enemy ships with the USS Missouri.
Down 200 enemies with artillery support calls.
Recover 5 ships.
Found all 28 hidden pegs.