Someone mentioned BBC Live Lounge before so I guess I'll add something as well:
I Believe in a Thing Called Love by Lemar is one of my favourite ones from it. Really chill and cool cover.
Also this is a jazzy instrumental rendition of Kanye West's Flash Lights by BADBADNOTGOOD.
Just gonna slip this in too: The Sun Ain't Gonna Shine Anymore by Keane.