Figured I'd just come in here and rave about Regina Spektor but...that's been done, so good on you. Here are the other things I had around the same level:
I'll be straight with you...when I'm alone and get drunk and nostalgic it's probably 50/50 that I post this on some music thread somewhere on the internet. Hope Sandoval is what all vocalists should aspire to be; her voice is heartbreakingly good and her performances convey an emotion that 99% of people I've seen couldn't muster on their best day. No matter what I'm doing, her performance here can always grab my attention.
It took me years to come around on this song. I'm a kid of the 90s. I like NIN. I hate country. This shit bugged the hell out of me. But one day it came on and it kind of hit me...this was a guy at the end his run and his vocals gave this such a punch that Reznor's version didn't have. This is one of the few songs that I just don't see the point to listening to the original anymore. Even Trent has admitted it isn't really his song anymore.
Kurt's vocals were the first that really struck me as someone being able to convey more from how they were singing than what the actual words were. I don't know that this is the best example, but it was one of the first that came to mind. Maybe it's hindsight, but especially on some of their acoustic stuff there always seemed to be a genuine sadness that didn't exist in the work of any of their contemporaries.
Godforbid is probably my favorite active vocalist. Dude has got a voice of gold. I don't think he has the genuine emotion of the above songs, but he makes it work.