This is a good question. Although it's a little tough to only pick 3 best and worse covers when there are so many. Instead of picking the best and worst, I'll give you 3 good ones and three bad ones, because if I was trying to figure out my 3 favorites, I could be here for days.
Good: Social Distortion "Ring of Fire" by Johnny Cash. Nuff said.
The Clash "Police and Thieves" by Junior Murvin. This song was the first bassline I ever learned to play.
Elvis Costello " What's So Funny 'Bout Peace, Love, and Understanding?" by Nick Lowe. A good song made truly great by one of the author's best friends.
Honrable Mention: Me First & The Gimmie Gimmies "Uptown Girl" by Billy Joel. Lots of fun.
Bad: Celine Dion "You Shook Me All Night Long" by AC/DC. The fact that she does air guitar while singing it, makes it all the more worse.
Nirvana "The Man Who Sold The World" by David Bowie. If you're going to cover a song, at least change it up a little, or what's the point? Kind of like Aerosmith doing "Come Together" as you so well pointed out earlier.
Smash Mouth "Why Can't We Be Friends" by War. Because, you just can't improve on perfection.
Had a lot of fun with this question. Thanks. Oh. and you're dead on with the American Pie thing. Madonna should be ashamed.