In agreement with Sarah and Dave about the Shatner version of Common People.
Don Ho doing Shock the Monkey is one of my favorite bad covers. It can be found on the truly strange comp "When Pigs Fly". It's not really funny as much as plain weird, with successful, respected (at least within their genre) artists covering songs completely inappropriate to their style. Like Billy Preston doing Girls on Film. Jackie Chan duets with Ani DiFranco on Unforgettable, which is actually sort of... forgettable.