Gen II
In Crystal, I was using Headbutt on a tree near the exit in Ilex Forest when a shiny Weedle appeared! I was so excited to capture it and raise it into a shiny Beedrill. I recently powered on Crystal again to see if it's still working. It was, so I nicknamed my Beedrill and entered him into the Bug Catching Contest here and won. :)
Gen III
In Emerald, I was darting through the route left of Lilycove when a shiny Poochyena appeared. I didn't want it to use Roar and flee, so I threw a Master Ball. A little less than a week later, I was darting through Pallet Town's route after beating the Elite Four and ran into a shiny Pidgey. I carried nothing weak enough for combat, so I again used a Master Ball.
Gen IV
In Pearl, I learned the chaining method, my first being a shiny Corsola. She has some of the most terrible IVs ever, but I love her color! I went on to capturing some 30+ shinies through the chaining method and then later RNG abusing more shiny Pokemon.
Gen V
In Black, which I had purchased at midnight launch, four hours later I reached Pinwheel Forest for the first time. I was very eager to capture a Cottonee. My very first encounter was a shiny female Cottonee. I was shaking so so so so so badly. It's one of my greatest Pokemon memories.
Gen VI
In Y, I finally learned the new rules in the chaining method. I was so nervous about breaking my chain in the 20s and 30s that I put the chaining on hold for hours. I finally captured my first Y shiny, a male Skiddo.