Laurie Anderson is a genius and all, but I'd classify her more as a storyteller than a classic singer. Anyway...
A word of warning: Do not start going through this thread when it's getting kinda late.
(Don't ask how I know. You can thank me later.)
Let's see whether we can't contribute a few more names here.
It seems Johnette Napolitano wasn't exactly a household name earlier in this thread. Shame because she's a great rock vocalist with quite some pipes, which she lends to
Concrete Blonde songs like this one:
Concrete Blonde - Heal It Up (Music Box,Hollywood 6/28/10)
Not bad at all considering this was 17 years on from when the song was recorded.
Staying in the realm of rock music but going back a bit further, those of you over the pond have probably not heard of Inga Rumpf and her soulful bluesy voice.
Frumpy - By The Way (live, 1972)
Speaking of things that are soulful, methinks the soul department is a little under-represented so far (which would be completely undeserved). So what about...
Minnie Riperton - Inside My Love
No wonder she was an inspiration to Kate Bush.
Or what about
Chaka Khan?
Another fine voice from the other side of the pond:
Linda Lewis - Spring song
This list probably is extremely incomplete even by the standards of extremely incomplete lists. Feel free to fill in the gaps.
Back in ol' Europe, let's head to
bella Italia with singer
Alice, who became known with pop songs such as
Per Elisa (1981) (which, in fact, is a reprise of Beethoven's
Für Elise)
...though I generally prefer her later output with high-profile international contributors, such as 1989's
Il sole nella pioggia, where this is from:
Alice - Le Baccanti
And finally, back to the UK for a bit of an oddball...
Danielle Dax - Inky Bloaters
Sometimes described as a "toned down version of
Diamanda Galás", she doesn't have the most ordinary voice but quite the range nonetheless. Her output ranges from experimental and cryptic to pop with biting political undertones, with all kinds of influences. A template for many modern-day indie artists.