The chance of hatching a random Shiny from an egg is very low and currently secret. This probability is also not static, since it changes from user to user, from day to day, and is also based on other factors.
The item Shiny Charm strongly increases your chance of hatching a random Shiny. You receive this key item after filling your third Badge set.
Shiny HuntItem Shop
A detailed explanation about shiny hunting can be found in the Pokeradar-article.
Route 53Honey, is very high.
You don't have to catch the Pokémon to see if it's shiny or not, since these special colored Pokémon have their own Artwork on Route 53.
- The Legendary Dog Trio (Entei, Raikou, Suicune) can be obtained as a shiny version at the Game Center Prize Exchange for a huge amount of Game Chips.
- The Lake Trio can be hatched as shinies randomly by redeeming Lake Trio Vouchers at the Game Center, although the chance is very small.
- A Shiny Lugia Voucher can be won at the Lottery if you purchased a lottery ticket containing all 5 matching Pokemon.