I never glitch when playing, but I had an unintentional one with Majora's Mask that was freaky (and no, this isn't gonna be the Ben drowned thing).
I think I jammed the cartridge into my N64 a little weird, then I turned it on and saw that I no longer had save files. Annoying, but since I was planning on starting a new game anyway it was no biggie. Go through the opening cutscene, skull kid leaves me behind, I gain control of Link annnnnnnnnd.... die instantly.
Game starts again, still in the woods area, and every c-button slot has the ocarina of time. B button slot where sword is supposed to be? Ocarina. Mask and item subscreens? Every slot filled with the ocarina. Press any button besides start and a? Link takes out the ocarina (but obviously doesn't know any songs).
To this day I regret not seeing how far I could get in this mode. Instead I panicked, turned off the system, took out the cartridge and put it back in... and suddenly all my old data was back. It was really weird!