I don't know about you but I find out of tune music hilarious. I believe the proper term is "discord" AKA music that lacks "harmony" AKA "music" that is painful to listen to.
The biggest example I can think of is Robo-Ky's theme from Guilty Gear XX, which is the discord version of Ky's theme (Holy Orders ~Be Just Or Be Dead~).
The only other example I have at the moment is Lunar Legend's "Blue Dragon's Song" (starts at 25:37) and its discord version (2:07:34). What the hell am I listening to.
Minor examples include Final Fantasy V's Piano Lesson side quest. In particular, the failed last note of Piano Lesson #5 never fails to bring a smile to my face. So close yet so far.
What is NOT suitable for this thread: e.g. Bare Knuckle 3's/Street of Rage 3's music. Sure, it's experimental and most people have a love/hate mentality to it, but the overall soundtrack was intended to sound good (and it mostly does). I'm only interested in music that was intended to sound bad.
Actually scratch that, I forgot about the four untitled "BGM" tracks from BK3. They're probably unnamed for a reason. Subjectivity aside, the rest of the music at the very least, have decent beats. I'm pretty sure Story of Thor/Beyond Oasis (also by Yuzo Koshiro) is also a candidate for the thread. For an action-RPG, the soundtrack has a severe sense of dread/uneasiness.
More to come if I come across them.