Shinies I've caught, and wasn't even trying to catch, they just appeared, what game, and where in the game:
Chinchou: Pokemon Gold (not HeartGold), fishing with Super Rod at Fast Ship port in Vermilion City. Status: Evolved to Lanturn, save file corrupted several months later after staying in my party with a level in the 70s.
Tentacool: Pokemon Silver (not SoulSilver), surfing in Olivine City fighting whatever Pokemon appear in order to level up. Status: Save file corrupted after never being taken out of the PC
Weedle: Pokemon FireRed, grass in Viridian Forest on my first time through. Status: Evolved to Beedrill, still in PC.