David Sylvian.
Bjork.
Sia Furler.
All 3 have some awful material, but all 3 sometimes get it just right 
This is going to sound nuts, but if the question really is about ‘unique’ voices that have me toe tapping, then I’ll say swagger and style do that more often for me than sheer vocal beauty. Steve Tyrell, yeah that’s right the big band guy with the gravelly voice, gets me every time. Ditto Rickie Lee Jones at her best: Under the Boardwalk from Girl at Her Volcano (man is that ever an under the radar album). How about Jersey Girl sung by Holly Cole on Temptation? So, there you go with a slightly different take on a fascinating thread.
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And then there is Serge Gainsbourg and Jane Birkin:
Je t’aime, moi non plus
La décadence
Love on the beat
Etc
Though not an all around favorite singer, the track “Before You Go” by Sachal Vasandani from the album Midnight Shelter. Been using it a lot as a reference to tell me things about systems lately as well. The recording has a nice palpability and that can be lost or skewed when something’s off in the chain.