This thread focused on sound and audio, not video or screenshot.

Let's start with some samples:

Tetris
Original Tetris:
https://www.youtube.com/watch?v=O0gAgQQHFcQ

Tetris - Game Boy - Version 1.0 (This verision only sold about 25,000 copies):
https://www.youtube.com/watch?v=Gz2qW4c3dzk&t=18

Tetris - Game Boy - Version 1.1 (Korobeiniki, aka "The Tetris Song"):
https://www.youtube.com/watch?v=whBg7IF5JUg

RealSound
- Enables 6-bit digitized PCM-audio playback on the PC speaker
Mean Streets
https://www.youtube.com/watch?v=havf3yw0qyw

Crime Wave
https://www.youtube.com/watch?v=MgDJ6EH-CBI

World Class Leader Board
https://www.youtube.com/watch?v=wivtSeGr_T8&t=9

Pokémon Yellow glitch music:
https://www.youtube.com/watch?v=m6mMSe_6hE8
(I like the Flight of the Beedrill selection.)

I find it interesting how the game's sound engine behaves when things don't quite go according to plans.

(Note: If you look for other glitch music, be aware that not all of it sounds decent; there are some harsh sounding ones out there.)

Metroid 2: Metroid Hatchling Theme
https://www.youtube.com/watch?v=fgxl2Kh8I04
https://www.youtube.com/watch?v=EtIScB_PXnE

This one is interesting in that different tracks of the music repeat after different lengths of time, resulting in a rather interesting effect (a few of the other late-game music tracks are like this). If you listen to it for a while, you will notice that it doesn't quite repeat exactly (it probably does eventually, but it takes a long time for that to happen). (Second link is to an extended version; I think it makes sense to include such a link due to the unusual nature of this music.)

There's also a game called Crazy Bus, which uses random notes for its music, and it sounds awful, so I won't link to it.

Cybernoid II - C64 soundtrack by Jeroen Tel.
Mach 3 - PCM audio over PC speaker.
Dune - AdLib Gold soundtrack by Stéphane Picq.
MegaRace - Also by Stéphane Picq. Cryo FM synth routines were awesome (I love that Maeva tune)...

...but not as good as the Vibrants Play Driver
"The Alibi" - It's not from a game, but it's absolutely stunning FM synth. EdLib conversion by JCH (Jens-Christian Huus), based on the original SID by LaXiTY (Thomas E. Peterson). See also "Gone" and "Gorgeous", both originally by DRAX (Thomas Mogensen) for C64 SID and converted to OPL2/3 EdLib format by JCH.

I don't know if it still exist, but the original original death shriek for ABUSE.

It was better than any horror film scream ever.

When the game first came out, it was on there as a wav and I copied it and put it on my old old PC to sound as a error.

Ooh, bookmarking. I'm obsessed with sound in general, so where sound meets video games I'm especially interested.

Machine-generated vocals that could play 10 hours (Crypt of the NecroDancer):
https://www.youtube.com/watch?v=I0xzDS7rcuc