Shinies are extremely valuable, even more than legendaries (in my opinion, anyway). The first shiny I was ever able to encounter was a Hoothoot in Silver, I think. If I remember correctly, I managed to catch it with only one Poké Ball, since its level was quite low. d: Shiny Gyarados was my first, technically, but I'm not going to count that, since I didn't encounter it by luck—it was due to an in-game event.
Another shiny that I was able to run into was when playing FireRed, which I think was an Onix. I was able to encounter it in Rock Tunnel while I was on my way to Lavender Town. When I first stumbled into it, I immediately thought that it was some sort of a new Pokémon, which surprised me a lot. But after examining it for a few seconds, I was able to figure that it was Onix. xD I wasn't able to capture it though, because unfortunately, I didn't have any Poké Balls with me at that time. If I did, then I would've caught it any heartbeat!
The most recent shiny I've encountered was a Litleo. I've tripped over this Pokémon at the beginning of Y on Route 12. When I did, I had a very limited amount of Poké Balls with me—only about three to five. I didn't want any of them to be wasted, so I thought that it'd be a good idea to weaken the Pokémon before capturing. When I landed an attack on it though, it became a critical hit, which made Litleo faint in an instant. I was quite upset about it, and if I wasn't being distracted by the TV, I probably would have cried. :( If I ended up capturing this Pokémon, then it definitely would've been one of my official team members.
Outside of the games, shines are just as rare. In the anime, the only time I've ever seen a shiny Pokémon was in one of the season two episodes. It was a shiny Rhyhorn, if I remember right. Also, when I was a kid, I used to collect a lot of Pokémon cards. I've had hundreds and thousands of them, but only two were actually shiny. xD Can't seem to recall what Pokémon they were, though.