When I read the thread title I thought of:
Johnny Cash
Tom Waits
Etta James
Leonard Cohen
Elvis
Leadbelly
Dolly Parton
Willie Nelson
Stevie Nicks
Their style, cadence, and timing has been copied so often that now what they were doing that was original and unique is just part of the genre. Also whoever was the first jazz-inspired pop singers to use those vocal flourishes that would be picked up by modern pop singers like Amy Winehouse deserve credit, I just don't know enough to cite a name.
For contemporary music, I'm sure there's plenty but I think of Colin Meloy from the Decemberists and the guy from the Mountain Goats in particular.