Shiny Pokemon can be found pretty much anywhere that Pokemon can be caught. They are, however, very, very rare to find. The chance of finding a shiny Pokemon without any increases to the chance is 1/4096. However, you can increase the chance of finding a shiny Pokemon. In all games after Black 2 and White 2, catching all of the Pokemon in the National Pokedex (excluding the event legendary Pokemon) will allow you to get the shiny charm. In X/Y, you can chain with the Poke Radar. (read more about Poke Radar chaining here) Another method that you can use in X/Y is chain fishing. (read more about chain fishing here) When chain fishing without the Shiny Charm, the chances of finding a shiny Pokemon can get up to 41/4096, which is about 1%. Yet another method that allows you to increase your shiny chance is simply by using the Friend Safari. In the Friend Safari, the chance of encountering a shiny Pokemon is 5/4096, which is, if I'm correct, five times higher with the base rate. If you have the Shiny Charm, the rate of encountering a shiny Pokemon is 7/4096.
In ORAS, you can DexNav chain. (read more about DexNav chaining here)
Also, in both X/Y and ORAS, you can use horde battles to find shiny Pokemon more easily. Hordes do not necessarily increase your shiny chance, but, as you are encountering five Pokemon at once, it allows you to find shiny Pokemon much more quickly.
In Sun/Moon and US/UM, you can use SOS chaining to increase your chances of finding shiny Pokemon. (read more about SOS chaining here)
In US/UM, Pokemon encountered while going on Ultra Warp Rides have an increased chance to be shiny. A chart describing distance and shiny chance can be found here on Bulbapedia.
Also, in all of these games, you can use the Masuda method. The Masuda method involved breeding two Pokemon from different countries. If you would like to understand and read more about it, go here.
Sources: This and this
