Use the following items at the Forge to upgrade the corresponding weapon.

Something Burnt: Galleon Mortar.
Something Coarse: Scrap Musket.
Something Fancy: Army Carbine.
Something Foul: Fire Bellows.
Something Greasy: Dueling Pistols.
Something Heavy: Cael Hammer.
Something Nasty: Fang Repeater.
Something Pointy: Brusher's Pike.
Something Sharp: War Machete.
Something Stringy: Breaker's Bow.
Something Wrong: Calamity Cannon.