I would argue (full-length essay idea...) that Xiu Xiu is one of the best, most creative and important bands of the 2000's. If you aren't up on the Xiu, think they're too weird or just haven't had time to get into them, I present a good place to start - what I think would be their greatest hits record.
"Crank Heart" - Fabulous Muscles (2004)
Had to start things off on an upbeat note. Great video game noises, percussion and a killer chorus make this one of the band's most accessible songs.
"I Broke Up (SJ)" - Knife Play (2002)
One of the best sing along spoken word parts ever ("This is the worst vacation ever...").
"Buzz Saw" - The Air Force (2006)
From the synth blasts to the odd percussion to the piano, one of the most prototypical Xiu Xiu songs.
"No friend Oh!" - Women As Lovers (2008)
Featuring prominent saxophone and lovely backing vocals from Caralee, surpasses "I Do What I Want..." as best single on Lovers.
"I Luv the Valley" - Fabulous Muscles (2004)
The band shows off the guitar in this one. Also manages to make a song involving suicide very listenable. Je t'aime the valley oh indeed.
"Save Me, Save Me" - The Air Force (2006)
One of the most busy Xiu Xiu tracks - filled with xylophone, horns, guitar and of course, synth, it also one of the most enjoyable.
"In Lust You Can Hear The Axe Fall" - Women As Lovers (2008)
Lovers features a lot of upright bass, most prominently on this track. Represents yet another turn the band has taken.
"Hello From Eau Claire (remix)" - Xiu Xiu: Remixed & Covered (2007)
While the original is great, this remix takes it to another level. The chessy techno back beat works, as do the ghostly interludes between Caralee's vocals.
"Muppet Face" - La Foret (2005)
If you don't feel something at the 1:13 mark, check your pulse.
"Ceremony" - Chapel of the Chimes EP (2003)
Stewart does one of his biggest influences right on this Joy Division/New Order cover. The practically screamed chorus and harder drums pushes this cover past the original.
"Apistat Commander" - A Promise (2003)
Jamie Stewart does some of his best vocal work here, tensely subdued to the point of explosion until the halfway mark.
"Fabulous Muscles" - Fabulous Muscles (2004)
One of the most starkly gross (lyrically) songs I've ever heard.
"Asleep" - Fag Patrol (2003)
This bare bones (acoustic guiatr/vocals only) song is the perfect closer.