I didn't even know what a shiny is until after I killed the red Gyarados, I was really noobish back then & thought that it had rabies or something & I have to kill it because it's a threat.
The only real shiny I caught was an Ursaring, I was grinding for days at Mt. Silver & suddenly found it, luckily I was on an emulator so I saved state just in case, I caught it easily since nearly all my team was over-leveled but it was boxed since then.
i had just entered victory road, just moved some boulders. a greeny/red golbat appeared. first time I've ever seen a random shiny. checked my pokeballs, only had my masterball. made me sad. but ohwell. needed that thing for ho-oh. now seeing red gyarados was one thing. i had caught him. but a golbat?! i had no idea..
In my Gold version, aside from Gyrados, I caught a shiny zubat outside of union cave at route 32. The sight itself was hilarious. I also caught a Golden Magikarp. Yes! Golden! I wanted to evolve the magikarp assuming it would evolve into a red gyrados and breed assuming I would get more golden magikarp but never got to trying it. Other than that, I cant remember any more for Gold.
In emerald version, I caught a green Golbat in victory road. I may have also caught a shiny mareep at the safari zone if I remember correctly.
Aside from Gyarados I've caught a Shiny Donphan, Weedle, and Oddish, and I've hatched a Shiny Elekid and a Shiny Magby from the ??? Egg in Crystal. Though out of all of them I only have the Shiny Elekid and the Shiny Magby now. One of them is currently stored in Pokemon Stadium 2 and the other one is on a rom on my phone.
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I had a shiny Golbat that I found on Victory Road. I didn't know that shiny Pokemon were ridiculously rare and I knocked it out on accident when I encountered it. Thinking it was like a Legendary Pokemon, I reloaded my last save and went back to look for it in the same area....and I encountered it again....and proceeded to knock it out a second time. I found it a third time and finally caught.
Haven't seen a shiny since then. Guess fortune is no longer shining on me. :P
like this guy, i found a shiny golbat on victory road. but i only had a masterball so i didn't even bother with it and ran. besides that, I've found a shiny raticate. but this was on an emulator.