When does mirage island appear?

It has a rare random chance of it being accessible. A number is generated between 0 and 65535. If the last 2 bit numbers matches your Pokemon's personality number, the island will appear on Route 130. This chance is much rarer than encountering a Shiny Pokemon. To know when the island has appeared when you are not on Route 130, you go into a house in Pacifidlog Town where a man is inside, then talk to him and he'll tell you whether the island is there or not.