Late to #ThursdayFiveList #UnderCover “Songs that someone else sang first”, but I can do what I want. 
IMHO good covers must almost completely alter the song such that it barely resembles the original and takes on new facets as a result.
1. H.I.M. - Wicked Game https://youtu.be/8oYodfK4DkE
Chris Isaak did a crooning mopey sensual ballad thing, H.I.M. turns it into a swirling mix of goth rock, glam metal, and yet still keeps that core crooning aspect. He sounds downright anguished as he belts it out.
2. Devo - (I Can't Get No) Satisfaction https://youtu.be/QmKQ2Z1odSc
The Rolling Stones version is full of sexual angst. Devo's version is so downright mechanical it feels like they're going through the bare minimum motions required to meet the criteria of "cover"... and yet at the same time this is the most DEVO sounding thing imaginable. Mocking the entire concept of rock n' roll while still participating in it.
3. Goldfinger - Rio https://youtu.be/PSCpWfWLMtM
Duran Duran's magnum opus- sped up, turned into a 3rd wave ska punk song, and then they take it a step further by throwing in some random references to The Monkees and Ronnie James Dio. Perfection.
4. Another Heaven - Running Up That Hill https://youtu.be/o4LHcXfs860
I'm not much of a fan of Kate Bush, but I readily admit this is a great song. Another Heaven takes it, slows it down into a distortion fuzzed out guitar riffed slog and renders it barely recognizable from the original. Kate's version feels like a pleading prayer with a glimmer of hope. This version feels more like that glimmer has turned to defeat.
5. Type O Negative - Cinnamon Girl https://youtu.be/BO9aD4mzSE8
Neil Young's version feels thin and almost whiny compared to the lush wall of sound that Type O brings to this track. Type O has such a distinct style that they can cover pretty much anything and it will cease to have any resemblance to the original. A genre unto themselves. (Though there are some immitators nowadays.)