If you are just encountering or breeding a wild Shinx, there is a 1/8192 chance of it being shiny. This applies to all pokemon that you just encounter. However, when breeding, you can use what's called the Mesuda Method, which is breeding a pokemon with a pokemon from a game of a different language (Like a shinx with a Japanese Ditto). This raises your chances of it being shiny to 1/2048.