One of my favourite glitches was in Hunter (1989) a 3D game where you are a commando. There are a variety of vehicles and creatures around the land, and whilst exiting a boat I somehow "boarded a seagull" and was able to fly around with my human model. Unfortunately I was unable to continue the game in any useful way, but it was quite an experience!
Another game from around 1990 was The Final Battle, a 2D isometric puzzle game where you have to excape from a medievil prison cell and click on the screen to interact with objects and people. There was one cave where if you dropped one of your items it would say "the shield has fallen to the floor. to the floor." and you were able to duplicate any object any number of times by dropping things and picking them back up (twice).
The original Carmageddon had some pretty funny physics glitches, sometimes when you drive into stuff you would get hurled across the map and your car would be so intensely damaged the polygons would form a hedgehog-shaped mess of triangles. I once blocked the train in Carmageddon 2 and it (despite it's infinite mass) somehow flipped on top of my and totally crushed my car.
GTA:IV and the swing of doom. Fantastic bug!
http://www.youtube.com/watch?v=DV_4ISH3nOsRunning Battlefield 1942 on an old slow computer and then reducing the AI to minimal CPU percent. I thoroughly enjoyed driving back and forth the main bridge in a jeep mowing down retarded Germans who were unable to use their weapons
The original Settlers game had a glitch were you could combine your roads with an enemies roads if there was a particular road layout, meaning you could steal their resources.
The orignal GTA - if you did doughnuts for long enough your car sprite would suddenly flip 90 degrees and you could drive your car sideways until your next collision.