A lot of terrific selections in this interesting thread started by Dawgbyte. Thanks everyone.
I would like to add the following, not as being "better" than the original, but certainly different and interesting:
Persuasions: Wanderer (U2 original)
Shonen Knife: Top of the World (from a terrific album of Carpenters covers called "If I were a Carpenter")
Cowboy Junkies: Thunder Road (Springsteen)
Nouvelle Vague: Guns of Brixton (Clash)
Black Box Recorder: Seasons in the Sun (Terry Jacks loosely interpreting Jacques Brel)
From curious to ridiculous, look for an album called "Golden Throats" it has a lot of the classic bad covers (Captain Kirk doing "Lucy in the Sky" and "Mr. Tambourine Man," Mr. French reciting "It Ain't Me Babe," Andy Griffith doing "House of the Rising Sun," and a whole lot more). On NPR I heard something really weird; it was Bob Dylan and Tiny Tim doing Sonny and Cher's "I Got You Babe."