The Sundays covering "Wild Horses"
To me these are some of the better covers of Stones songs out there. The Stones, like the Beatles and every other mega-band, have been covered to death by everyone under the sun: Some with very nice voices, some who ought never open their mouths in the presence of others; some who play it safe with a note-for-note copy of the original and some who distort it so much (Cat Power, Stina Nordenstam) that it really is a brand new song (sometimes very good on its own, sometimes...not so much).
I like these two covers because A) I like both the original songs very much anyway, B) I loved both Terence Trent D'Arby and The Sundays long before they ever played these songs, and C) The covers are in my opinion excellently done, neither too exact nor too great a departure. One is intense and stripped down instrumentally, and one is more melodic, but both (I feel) are moving in their own way, the singers both putting their hearts into it as much as they do their own material, and thus they've made a nice tribute to the original songs and songwriters.
I generally don't care for covers, but like everything in life, there are always exceptions. Do you have some favorite covers? Not just of the Stones, but cover versions of any songs that you especially like? Feel free to name them in the comments section!