Loathen is a silver sword in The Witcher 3: Wild Hunt, which can only be found as random loot.
It is more likely to spawn in cave locations in Velen and Skellige, outside of which it only appears at guarded treasures.
One place where this sword spawns frequently, though not guaranteed, is the guarded treasure (Noonwraith) in the mine cave just south of Frischlow, where Geralt can go for the gauntlets diagram during Scavenger Hunt: Griffin School Gear/Part 2.
Another location where it is known to appear is Yustianna's Grotto, in a chest on the platform exactly where the bandits during Yustianna Disturbed are.