First you need a Pokemon that has the ability either Sticky Hold or Suction Cups and put in in front of your party.
Register the Good Rod (reccomended) because it gives you a high rate to catch a shiny and unregister everything else.
Pick a spot where you want to fish and and hit Y to cast your rod. Suction Cups and Sticky Hold will garantee that you will get an encounter every single time. The same, don't run away from them. It's easier because you don't have to move. Stay in the same spot. Eventually you will get a shiny fish. Some people find them after 4 fish, and some of them find one after 140.
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The 3 Ways to Get a Shiny Pokemon in Pokemon X/Y
Random(C) to the owner. If you guys want a shiny Pokemon, there are three ways to get one: Masuda Method, PokeRadar and Chain Fishing. In my opinion I'll recommend the Masuda Method, because you can get one of your favorites. There's a video too on how to...