I am not saying these are "better" than originals, but, I like the following:
Be Good Tanyas "Waiting Around to Die" (Townes Van Zandt)
Cowboy Junkies "Thunder Road" (Springsteen)
Shonen Knife "Top of the World" (Richard Carpenter)
Nouvelle Vague "Guns of Brixton" (The Clash)
Sid Vicious "My Way" (Thibault/Anka)
REM "First We Take Manhattan" (Leonard Cohen)
Concrete Blonde "Everybody Knows" (Leonard Cohen)
Nazareth "Love Hurts" (Roy Orbison)
Diana Krall "A Taste of You" (Joni Mitchell)