Instructions
Stock up on Pokeballs in advance. Always have plenty with you, as you may run into a Shiny when you least expect it.
Teach Hypnosis, Sing or Yawn to one Pokemon on your team. You'll want to keep this Pokemon with you at all times so you can put the Shiny Pokemon to sleep. Putting a Pokemon to sleep raises the capture rate, making them easier to catch.
Lower the Shiny Pokemon's health into the red, but take care not to knock it out. Teach one member of your team the move False Swipe, which reduces an enemy Pokemon's health to 1 HP without knocking it out.