I've only encountered 4 shinies throughout the games, and I've generally had some bad luck with them.
-The first I saw was a shiny Teddiursa on my Pokemon Gold. I ended up getting a one-hit KO thanks to a critical hit, and I've seen very few Teddiursa (let alone shiny) since then.
-The next one I found was a shiny Shroomish. This is the only one I've actually caught, but I didn't even realize it until a year or two later when I was cleaning out my boxes on Sapphire. I must have caught it just for the Pokedex, and never even realized it was shiny.
-The third I encountered was a shiny Paras in the Safari Zone on Fire Red. I was fooling around and wanted to see if I could catch a Chansey with only 1 step and 1 Safari Ball left. With my luck, I ran into a shiny Paras, and I didn't want to risk it running away, so I threw my Safari Ball at it right away, with no luck.
-The last one I found a few weeks later was when I was playing Fire Red on VBA. I was just randomly going through it when I was stopped by the old man who teaches you how to catch Pokemon. A Shiny Weedle appeared, and I was quite mad at the fact that I found a shiny on the VBA, and I wasn't able to even catch it!