I think I should reign this thread in a bit. Definitely many unique and original voices here, and a whole lot of talent for sure. I was thinking more along the lines of an "ugly" voice, for lack of a better way of putting it. A voice you'd otherwise never think of as a singing voice, or a musical voice even. Like Howard suggests, a voice that if you only heard a few bars you might think someone was pulling your leg and had played something from a nightmare karaoke session, yet if you take the time to listen to the whole song, or contemplate an album or larger body of work you may really be impressed. Lets see if I can put this back on track a little bit...
Paolo Conte...much like Tom Waits his voice sounds ravaged by drink and cigarettes.
A few from above that I'd connect with that way. Dylan, Louis Armstrong, Marianne Faithfull, Yoko Ono.
A few from above that I wouldn't think fit the description: Casandra Wilson (damn, her voice is like liquid honey...albeit my wife finds her a bit 'manly'...I'd say 'unique' for sure, but definitely not "ugly"). Allison Krauss, Gillian Welch, Aaron Neville, and others - not to pick on anyone specifically, but all of these voices are indeed "unique", but none are what I'd call an "ugly" voice. Perhaps the title of the thread is misleading (my bad). I should have titled it "An Ugly Voice". Interesting contributions all, regardless. Keep'em coming.
Marco