I was going to post a list of my favorite covers, but that seemed too daunting. Instead, I'll just highlight one particular cover--Lick the Tins' rendition of Elvis Presley's "Can't Help Falling in Love"--that is at least equal if not better than the original version. This track can be found on Lick the Tins' 1986 album, Blind Man on a Flying Horse, and on the soundtrack to the 1987 John Hughes film, Some Kind of Wonderful.
As an aside, the Some Kind of Wonderful soundtrack is worth getting for the other songs as well. Included is a slightly peppier (and more satisfying) version of "The Hardest Walk" by The Jesus and Mary Chain (compared to the version found on Psychocandy) as well as a very '80s sounding cover of The Rolling Stones' "Miss Amanda Jones" by The March Violets.