My first ever shiny showed up on LeafGreen, about eleven years ago. I was trying to catch Zapdos, and had to keep resetting because it would either wipe my team or drain my bag of Poké Balls. On the sixth reset, I saw sparkles. I wasn't sure if I was imagining it or not, because Zapdos's shiny form is so similar to its normal form; but I managed to catch it, and sure enough, it was shiny. That Zapdos is sitting on Omega Ruby at the moment.
My favourite shiny is probably Absol; it just looks really cool, with those bright aqua eyes. It probably also has something to do with the fact that I managed to get a shiny one using the Masuda Method, after what must have been over 900 eggs.