Excluding the shiny red Gyarados and hatching a shiny egg Pokemon in Generation II (which I shouldn't be mentioning in this section) I have seen and captured several shiny Pokemon:
Spinda- this happen as I ran out of Max Repels because I was trying to get a lot of ash in order to build furniture for my secret base and I just ran into a shiny one and captured it; I released the original colored one right after. This was in Pokemon Emerald
Zigzagoon- this happen in my Pokemon Sapphire version. I was strolling through Petalburg woods trying to hatch an egg and I ran into a shiny Zigzagoon by accident because I thought I had a Max Repel on.
Mawile- this happen in my Pokemon Ruby version. I was searching for a Mawile in general in Granite Cave and after searching for about 10 minutes, I found one
Tentacool- this also happen in my Pokemon Ruby version. I don't remember when, but when I was surfing to go to another town, I thought I had a Max Repel on, but I did not have one and then I just ran into a shiny Tentacool.