In my day, I've experienced some glitches while playing games, and these are those that I remember.

In both the Dreamcast and PS2 versions of SF3: Third Strike, if you do a super after killing an opponent i.e. within a combo and the hit before the super kills them, your super meter in the second round will still be full!

In the SNES Secret of Mana, you can get stuck in plenty of "blackspace" when doing the whip-jump over chasms. I got stuck in the Mana Fortress and had to reset, because I was completely immobile! NPC's constantly get stuck behind stuff and will have to teleport to get to you.

While playing Prince of Persia: Warrior Within, the sound got stuck for a stupid switch and kept replaying constantly. I had to reset the system to fix it. Also, there are some weird trigger event hiccups; like when spikes are supposed to shoot up where you stand, if you go back in time, they remain up....forever, thus blocking some necessary jumps and totally screwing you from moving onward. Overall poorly debugged.

After playing Valkryie Profile for a while, I noticed a weird bug that I could replicate. When you have both Grey and Valkyrie in your party, and do Grey's Icicle Disaster, just before he hits the big iceberg, press the Square button as if you could do Valkyrie's Niebelung Valesti. Grey won't say anything...then Valkrie will do her final move thing, and immediately after she says "Neibelung Valesti", Grey goes "Icicle Disaster". It was amusing.

While playing FFX, I was running around Mushroom Rock Road and was close to the edge of the grass to the left of the path. Well, I was messing around running at that little bald patch in the middle, when I got in a battle, after the battle, I kept running to the left, and after like 30 seconds, I went through the 'invisible wall' and started running through the freaking background. There were no battles, I was just running through the map. I went through all objects and couldn't get back onto the map. So I ran way off into the background until Tidus was a speck, and then all the way into the camera, then I reappeared on the path! And it worked like normal after that. I have Marcy as a witness to this event too.