My favorite glitch was the Link's Awakening SELECT screen glitch. You could basically teleport anywhere using this glitch, even access areas you aren't supposed to access yet, and technically trash your gamefile using it )
Another glitch I enjoyed IMMENSELY (and I mean REALLY REALLY A LOT) was the Missingno. glitch for Pokemon R/B - I spent a lot of hours doing it, seeing how much I could mess up my game, and I really really loved walking around the "glitch city" that would occur if you did it the Safari Zone way. For some reason, I really thought I could find Mew there, which is why I kept returning.
Well I have to say that when I think of glitches I tend to think of the ones that frustrate the heck out of me and usually happen at the worst possible times. I certainly have my fair share of examples for those. I am glad to see people using them as a positive, though, such as @Karasu above. This really makes me take a different approach towards them and now I want to see if I can get one of those good ones. Thanks for sharing.
My favorite game glitches of any kind were always in San Andreas. I like seeing random glitches when it's not annoying.
I had a glitch one time when I was on a killing spree, and all of a sudden when I hit 5 stars, as I was firing my weapons, they started shooting faster, and my bullets were unlimited for some reason. Great glitch when you have cops coming from all directions.
I think it was because I input cheats very fast as I play.