In every main series Pokemon game besides Red, Green, and Blue, you can find a shiny normally in the wild with a 1/8192 chance on your side.
However, there are exceptions to this rule in a few selected games.
Shinies sprites did not exist yet, however, the coding for shinies did, at least in Yellow.
-Here is a video of the glitch to get a shiny in Yellow: http://www.youtube.com/watch?v=V0QDZQhjxzY
To see the shiny sprite, transfer it to a gen 2 game.
Diamond, Pearl, Platinum:
1) Complete the Sinnoh Pokedex, then talk to Prof. Rowan. He will give you the Pokeradar. With enough patience, you can get almost any shiny with it.
-Tutorial on how to use it: http://www.youtube.com/watch?v=dkeKPlBYYbs
2) -Get a Pokemon (preferably ditto) from a different language as yours.
-Breed it with a Pokemon of yours
(works in every game after)
Pokemon White 2 and Black 2:
1) Complete the National Pokedex and talk to Prof. Juniper's dad. He will give you the shining charm. It increases your odds of finding a shiny in the wild.