I'm possibly cheating because this Russian-born American
musician has a strong cult following. I'm talking about Regina
Spektor.

People have called her music everything from anti-folk to
what-Tori-Amos-used-to-be. She employs Eastern European and Jewish
elements and techniques (how often do you hear the glottal stop
used in pop music?).
She's a classically-trained pianist so her melodies often sound
like they could came out of an early 1900s concerto. She also has
some guitar songs, and how can one voice take on so many textures
in different songs - haunting, gritty, gentle and more.
Her early work (in the 1st 3 albums) was "dry" in production,
with next-to-no vocal reverb. If you watched 500 Days of Summer,
the song at the beginning ('Us') is a weak example of that. The
last 2 albums (and the most popular to date) are less raw, whether
that's good is debatable.
The best thing about her is that she can make up complete songs
on the fly (one of them's below), so you're never exactly sure
what's on her setlist at live gigs. And her wit still shows through
in the lyrics of songs like 'Us', 'Hotel Song'.
Albums: 11:11 (2000), Songs (2002), Soviet Kitsch (2004), Begin
To Hope (2006), Far (2009), What We Saw From The Cheap Seats
(coming 2012)
Us (live - Christ, watch her fingers!)
Braille (one of the "dry" songs)
Reginasaurus (live and impromptu in '04)
Man of a Thousand Faces (music video)