Randy Newman's original
Three Dog Night's cover
Pretty much all of Three Dog Night's big songs are covers (Mama Told Me (Not to Come), Eli's Coming, Liar, Joy to the World, Never Been To Spain, Sure As I'm Sittin' Here, The Show Must Go On, and One), but this one is my favorite.

Hooked on a Feeling:
B.J. Thomas's original
Blue Swede's cover
Not that the original is bad (I prefer it at times), but the cover is awesome itself.
Flying:
The Beatles's original
Secret Machines's cover
I like most of the covers from Across the Universe, but this one is my favorite (with Eddie Izzard's rendition of "Being for the Benefit of Mr. Kite!" in a close second).
This isn't close to being all of them, but it's a good start for anyone interested.