There are 5 swords that you can find in the game. Swords are customizable, they can be sharpened so they inflict more damage and they can have an added status effect to them if you visit a blacksmith.

The sword you start out with. It has 5 damage points initially.

Can be bought in Savina for 35 Pix by Doleos in the Savina marketplace. You get to try it out before you buy it in Savina, in the sidequest "Helping Jaden". It has 15 damage points.

True Flame
Can be found in the Thieves' Passage in a locked room, along with 28 Pix. It has 22 damage points.

Twin Serpent Blade
Will be given to you if you survive the Summoner's Challenge. It has 30 damage points.

Can be found in Blackwhisper Cave after completing the game. It has 45 damage points.

Can be found at the foot of the Mogwai Mountains after completing the game on the hardest difficulty. It has 190 damage points.