I nominate J. Cash's cover of "The Mercy Seat" a cover of Nick Cave And The Bad Seeds from the 1988 release Tender Prey. On the lighter side, Any cover that I've heard by Big Head Todd And The Monsters, or Lake Street Dive has been excellent (both BHTATM, and Lake Street Dive get mucho play at my house, most of the catalog is on CD)...
my little garage punk band did Leonard Cohen's "Teachers" my favorite, since I got to croak it.