Shiny Hunting
Shiny hunting refers to the act of attempting to hatch a specific Shiny by hatching the same type of Egg repeatedly. This can be done with no upgrades by trying for a shiny with a 1 in 1024 chance, that's about a 0.1% chance of a Pokemon randomly hatching as a Shiny. You cannot chain a certain Pokémon without upgrades, so hatching the same kind of Egg in a row does not increase the chances of finding a Shiny in this case. Users without Hypermode and a Radar who are on a hunt may hatch as many different kinds of Eggs in any order as they want.
There are special circumstances as to when everyone gets a shiny boost.
To further increase your chances of hatching a Shiny, you'll need the PokéRadar and be in Hypermode to chain a certain Pokémon. The PokéRadar alone lets you chain up to one Pokémon, raising your chances quite a bit. With Hypermode, you can reach the maximum capacity after consecutively hatching 40 Eggs of the same Pokémon species. This boosts your chances up to 3,996%, about a 4% chance a Pokémon will hatch Shiny. At this point, hatching more of the same Egg will not increase you chances, but will result in hatching a Shiny sooner or later. This also has a lot to do with luck, though, so don't give up soon after reaching max chance! It might take hundreds of Eggs before you hatch a shiny - or it might hatch very early in the chain.
Further boosts can be obtained through a Shiny Charm, which doubles your boost for a full 24 hours. Shiny charms disappear after 24 hours, unless the user owns a Shiny Charm Box. However, the Shiny Charm will still need recharging to be used again. Shiny Charms can be bought and recharged with Gold Poké.
All boosts stack with the Pokémon of the Day and Sei's Counter.
There are some Pokemon that can not yet be chained, at least not easily. These include Pumpkaboo, and the Unown forms, due to the egg sprite and name being the same for the different forms.
