One that I was surprised to learn that I like to share is Grace Jones 'Private Life' was originally by The Pretenders.
Hynde is quoted as saying: Like all the other London punks, I wanted to do reggae, and I wrote "Private Life". When I first heard Grace's version I thought 'Now that's how it's supposed to sound!' In fact it was one of the highpoints of my career - what with Sly and Robbie being the masters, and Grace Jones with her scorching delivery.
Not really a cover but it used such a huge chunk that it pretty much is...
Beyonce "Crazy In Love" is basically The Chi-lites "Are You My Woman"