The following swords are listed by order of strength. Complete the indicated task to unlock the corresponding sword.

Cutlass: Default sword
Dutchman sword: Complete the Pearl Vs Dutchman level.
Norrington sword: Complete the Isla Cruces level.
Bone sword: Complete the Davy Jones' Locker level.
Gun sword: Complete the Sea Battle level.
Pirate King's sword: Complete the Shipwreck City level.
Davy Jones' sword: Complete the Maelstrom level.
Ghost sword: Complete the Hit Combo sub-mission on the Isla Cruces level.
Legend sword: Complete the Hit Combo sub-mission on the Maelstrom level.