The lyrics of Billy, Boney and Ma may be strange, the music hardly could be more conventional.
My nomination: Design by Gentle Giant. Two beautiful melodies which absolutely have nothing to do with each other and create an ugly effect only very few people can endure. I'm one of those even rarer people who love it.
http://www.youtube.com/watch?v=D22xWl9PTwUBut for really weird music you have to turn to modern classical music. Two of my favourite weird compositions are
Nikolai Roslavetz, Second Sonata for Violin and Piano.
This music is completely atonal, but unlike Schönberg, usually Berg and always Weber Roslavetz also manages to write elegant and melodic music. Every time I hear it I fall under its spell.
Galina Ustvolskaya, Fifth Pianosonata
Almost 18 minutes of variations that centre around one tone, a D. It's from 1986. The piano is treated like a percussion instrument. The listener has to be very tolerant towards dissonants. Still it creates a forceful impression that only can be described as spiritual. For this it has to be played as loud as any metalsong.
There is nothing even remotely like this Fifth Pianosonata, except a few works of this Russian lady herself.
If anyone thinks Frank Zappa or Captain Beefheart avant garde, try this. Preferably try to find Dutch pianist Reinbert de Leeuw performing it.
I'll be curious if anyone here can get through even the first 20 seconds.