I second Imogen Heap! You can actually hear her entire new album, "Ellipse" for free over at this handy-dandy link:
http://www.imogenheap.com/ellipse/

Songs I would suggest are "First Train Home", "Wait It Out", "Tidal", "Canvas" and "Half Life".
As a cool piece of trivia, the track "Earth" was recorded without any instruments whatsoever. Totally a Capella. The only sounds you hear are Imogen's voice, her beat boxing, and a bit of vocal distortion.

Also, here's the music video for her song "Goodnight and Go", from her previous album, "Speak for Yourself":
http://www.youtube.com/watch?v=qutKFVLj2A8

I'm also a huge fan of Amanda Palmer, the front woman and one-half of the band The Dresden Dolls. You can hear all her solo music over at this site:
http://whokilledamandapalmer.com/

Just click on "The Album" and choose the songs you want to hear. A player should automatically pop up on the next page, along with the lyrics. I suggest "Astronaut", "Ampersand", "Leeds United", "Guitar Hero", "Have to Drive" and "The Point of It All".
You can also see her fantastic video series for most of the songs on the album under "The Videos". If those are loading too slowly, here's her official YouTube where you can find them all:
http://www.youtube.com/user/amandapalmer
Here's the first part:
http://www.youtube.com/watch?v=YmwVH2YjgU4&feature=player_profilepage
The rest should be under "More From: Amanda Palmer". :3

There's also Joanna Newsom. She plays the harp. Some people don't like her voice, and some people take a while to get used to it. I really love it personally, but you can find out for yourself. You can listen to samples of her music on her official Drag City Records page over here:
http://www.joanna-newsom.com/

Scroll down to where all her albums are. You can play samples from "The Milk-Eyed Mender" and "Ys", her two full-length records. Additionally, here's the music video for "The Sprout and the Bean" from her first album:
http://www.youtube.com/watch?v=IYl0uLrXP7U&feature=related

What else...
I really love Kate Nash and She & Him.

Kate Nash- Foundations: http://www.youtube.com/watch?v=uhC4E055OsY

She & Him- Why Do You Let Me Stay Here?- http://www.youtube.com/watch?v=FkzRyHa9a6g
(Yes, if you're wondering, that is Zooey Deschanel. :3)

Well, I think that's all for now. Gosh, no guys. I should get into more male performers. I'm surprised I actually can't think of any I really like. xD
Hope you like some of this stuff.

Shuri/Eli