I kinda like a bit of Cartoon Rave music from time to time, some are kinda of embarrassing to be caught listening to, but fun all the same:

youtu.be/PMrjgD9PCnk - Bolt by Horsepower
youtu.be/cQsYXG1ONaM - Rubarb & Custard by Shaft
youtu.be/odQgfUUegQI - Sesame's Treet by Smart E's
youtu.be/6O6TqhrAcVo - Trip to Trumpton by Urban Hype

Enigma is one of my favorite bands (well its not really a band, but a sound mixer), and I always get the weirdest looks when I'm playing it in my car. I have a thing for gregorian chant beats.

Terrorcore

www.youtube.com/watch?v=np0kau…

Of the music I listen to the most the Revolting Cocks are the oddballs of the bunch.  They did covers for Olivia Newton John's "Physical" and Rod Stewart's "Da Ya Think I'm Sexy?" which are awesome.

www.youtube.com/watch?v=jGIy6b…

www.youtube.com/watch?v=VOq_D-…

Dude, everything I listen to is weird but I love it. Here's just a few picks though.
www.youtube.com/watch?v=vMSQ0J… The Flashbulb - Chik Habit
www.youtube.com/watch?v=VPz1Up… The A.M.D.A - U R Weird M8
www.youtube.com/watch?v=BmpFXN… hyi - she spoke of it
www.youtube.com/watch?v=UOEavj… The Quick Brown Fox - Year of The Bad Dragon

In terms of 'weird' songs I like these two: Lego Racers - Menu and Lego Racers - Build Mode. They are both quite jumpy and goofy.

I'm loving this song a lot lately

www.youtube.com/watch?v=Ry_f2o…

For some reason I really like this.

www.youtube.com/watch?v=2dllo8…

My favorite band is Oingo Boingo. They're not very popular around where I am. 

I like all kinds of music, which makes me the weirdo with all types of fans
I've experienced it with all my friends that likes different music. Many of my closest friends are really into hip hop, rap and mainstream music, when i show them for example Helloween, Ramones or InFlames or something other of the like, they're like "naah, its really not my thing" or "how can you listen to that?".

Same thing goes for other friends that like metal and different kinds of rock and when i show them some electro or hip hop or rap or something, they're like "aw thats garbage" or "that not real music".

I appreciate alot of different genres and its makes me a little bit sad that people dont want to enjoy everything, but then again, you do you bobo  

I really love the Irish-Canadian folk band The Irish Descendants

Come Out ye Black & Tans, Step it Out Mary, and Barrett's Privateers are all magnificent works by them. 

TV show themes

Star Wars themes.
It makes every journey to a different website feel oddly epic.
Lmao.

Hmmm... There'd be a few candidates out of my whole selection that I think are slightly weird.

Rap:
Twista - Celebrity Overnight

Reason I think it's weird is because the beat is mostly made by a violin. It's unique, and weird in a way but it still sounds good.

Rock:
Chronic Future - Time And Time Again

Reason I think it's weird is because the band tried a weird mix of punk rock and hip hop with this one, but it turned out well none-the-less.

EDM:
Virtual Riot x 12 Planet - Leave It Behind (Ft. Ash Riser)

This is supposed to be a dubstep track. And it is. But that swing/jazz introduction threw me off for quite a moment. I like that weirdly unique throwback to classic music.

This is everything my dreams are made of. 
www.youtube.com/watch?v=EdooYa…

Does King Missile fit into this category?

Mac Sabbath-a McDonalds themed cover band that edits the lyrics of Black Sabbath songs to correspond to their fast food related theme
and The Radioactive Chicken Heads-not real a big favorite of mine and describing them is a hard task, i'd recommend you look them up to get a feel for what they're like