Note: The following swords are listed by order of strength. Successfully complete the indicated task to unlock the corresponding sword:

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