Most unusual/mesmerizing female voice.

My current favorite singer, possessing an immediately recognizable voice, is Iris Dement. There is a clip on You Tube of Emmylou Harris harmonizing with her. Iris is a great songwriter as well as singer.

Chante Moore

I find Kat Edmunson's voice particularly intriguing.  Very old school jazz sounding, but she uses contemporary arrangements and I like the combination.  But I'd like the voice regardless.

Emma Kirkby
Eva Cassidy
Youn Sun Nah
Cesaria Evora

Lisa Gerrard
K.Bush
Bjork
Diamanda Galas

I may add some italian female singers, like Alice and Elisa, but I guess they're not so well known in the USA

but the one that mesmerizes me most, though not so unusual:
Kate Pearson
her voice is not "uncommon" but she definitively touches something inside me, her voice's vibrations cause resonances in me
I've read she now has retired and manages a hotel somewhere in a lonely place in the mountains; well, when I come to USA, no NY or LA, no Vegas, no Yosemite or Monument Valley: I'll go in that lost place and ask her to sing... :-D