There are 13 Trophies that can be earned in this title.

Bronze

Cleared Stormy Sea
Clear the Stormy Sea in Arcade.

Cleared Mountain Stream
Clear the Mountain Stream.

Cleared Cave
Clear the Cave in Arcade.

Cleared Great Whirlpool
Clear the Great Whirlpool in Arcade.

Cleared Treasure
Clear the Treasure in Arcade.

A New Beginning
Register a score in Ranking Mode.

Synchronized Shooting
Reach a Teamwork Level of 100%.

Davy Jones' Locker
Reach a total of 10000 kills.

And a Bottle of Rum
Obtain a YO-HO-HO rating while manning the helm.

Silver

Navigator of the High Seas
Obtain an S rank in all stages.

Captain of the Seas
Clear all stages in Ranking Mode from Stormy Sea onward.

Sharpshooter
Obtain a 'Nice Combination!' rating 100 times.

Gold

Ruler of the Seven Seas
Clear all stages in Arcade Mode from Stormy Sea onward without continues

http://www.ps3trophies.org/game/deadstorm-pirates/trophies/