Here is the full list of all 15 Battleship trophies - there are 13 bronze, 1 silver and 1 gold.
Complete all Tutorial and Campaign missions.
Sink a total of 150 ships.
Complete the Campaign's first mission.
Play 5 games with the pirate fleet.
Play 5 games with the tech fleet.
Play 5 games with the orc fleet.
Play 10 games with the military fleet.
Successfully hit a total of 125 coordinates.
Detect a total of 50 ships' coordinates before hitting them.
Reveal all of your opponent's grid coordinates before winning a VS game.
Play a local Player VS Player game.
Play a game using Classic Rules.
Win a game in any VS mode.
Sink your first enemy ship.
Successfully hit 10 consecutive ship's coordinates.