Due to the length of this answer, it will be split into two parts. This section contains an introduction to shiny hunting, along with some of the odds-boosting methods. The other section contains the rest of the methods not covered in the first section. To jump to section two, click here.
Ever since their introduction to the series, Shiny Pokemon have been some of the most valuable trophies you can obtain. They’re incredibly rare, however: the odds are so low, you can go through hundreds of hours of gameplay and never find any. To make the process of finding a shiny (dubbed “shiny hunting”) less tedious, each game has introduced different methods to increase the chances of finding a shiny. All make it easier to find an off-coloured Pokemon, but the rates vary by method and game.
For simplicity’s sake, it will be noted here that from Generations II to V, the base shiny odds were 1/8192, or 0.01%. With the release of Generation VI, base shiny odds were doubled to 1/4096, or 0.02%.
Wild encounters
The most obvious (and arguably most difficult) way to find a shiny. Every time you battle a wild Pokemon, there is a 1/8192 chance (approximately 0.01%) prior to Generation VI that the Pokemon will be shiny. With the advent of X and Y, those odds were doubled to 1/4096 (0.02%), making it a bit easier to find shinies. There are exceptions and modifiers that can be implemented to make it easier to find shinies in the wild, which are covered later.
It should be noted that some Pokemon are shiny locked, meaning they can never be shiny, no matter how many times you look. This mostly pertains to notable story Pokemon like Zekrom and Reshiram in Unova, or the Galarian starters at the start of Sword and Shield.
Breeding
Breeding two compatible parents will produce an Egg that has a chance of being shiny. In most cases, this has the base odds of whatever generation you’re playing in. There are ways to boost your chances, though:
Introduced in Generation IV, the Masuda Method significantly increases the chance you get a shiny if the proper conditions are in place. Two compatible parents of different nationalities must be placed in the Daycare (for example, an English Pikachu and a Japanese Ditto). The resulting offspring will have a far higher chance of being shiny, with odds of 1/1638 in Generation IV, 1/1365 in Generation V, and 1/683 from Generation VI onwards. The Masuda Method can be stacked with the Shiny Charm from Generation V onwards in order to produce even higher odds.
In Generation II only, shiny mechanics worked differently, being based on IVs. Because of this, if the parent is shiny and of the opposite gender of the offspring, the latter has as high as a 1/64 chance of being shiny. This only works in Generation II, and only if one of the parents are shiny.
The Odd Egg in Crystal can hatch into one of seven Baby Pokemon (Pichu, Cleffa, Igglybuff, Tyrogue, Smoochum, Elekid, Magby) and in international releases has a 14% chance of containing a shiny Pokemon. In Japanese editions of Crystal, the Pokemon that hatches as a 50% chance of being shiny. You get this Egg as a gift from the old man in the Daycare Center on Route 34.
Pokemon in Generation I can never be shiny, but if you transfer them to Generation II, there’s a chance. Due to shininess being determined by IVs, it’s possible to transfer a Generation I Pokemon to Generation II via the Time Capsule and find that it’s shiny. Only fishing encounters, stationary Pokemon, Gift Pokemon, or in-game trades have a chance of being shiny when transferred.
Guaranteed shinies
There are a few shiny Pokemon that you can obtain throughout the course of the game you’re playing. There aren’t many of them, and even fewer are easy to obtain.
In all Johto games (GSC, HGSS), a shiny Gyarados waits at the Lake of Rage. You must defeat or capture it to progress with the story. It won’t respawn if you knock it out.
In Black 2 and White 2, if you see every non-mythical Pokemon in the Unova Pokedex, you will be granted access to the Nature Preserve. In the middle of this area resides a shiny Haxorus, available for you to battle and catch whenever you choose.
Black and White 2 also hold two other guaranteed shinies: Gible and Dratini. Each is a reward for defeating Benga in the postgame battle facility of your game — beating him in Black 2 will give you the Gible, while his defeat in White 2 will reward you with the Dratini.
In Legends: Arceus, there’s a special research quest available for a shiny Ponyta. It can be found very early in the game, and is consequently one of the easiest shiny Pokemon in the series to obtain.
Select event distributions are also a way to pick up guaranteed shinies. Occasionally, Mystery Gifts have been (and will continue to be) distributed to players, rewarding them with shiny Pokemon. The shiny Zeraora in Pokemon HOME is one of the more recent notable examples of such an event.
Poke Radar
The Poke Radar makes appearances in Diamond, Pearl, Platinum, X, Y, Brilliant Diamond, and Shining Pearl. It can be obtained after you receive the National Dex, and significantly boosts shiny odds. Activating the Radar in a patch of tall grass will cause other patches to shake. Capturing or knocking out the Pokemon in that grass will start a chain, which can be continued by capturing or knocking out Pokemon of that same species with the Poke Radar. At the highest chain of 40, shiny odds in DPPt are around 0.5% (or 1/200) per patch of grass, and it’s generally accepted that XY have the same odds. The BDSP odds are higher -- a chain of 40 yields 1/99 odds, or around 1%. For a more comprehensive understanding of the Poke Radar's mechanics in BDSP, Serebii's guide is a good place to look.
Cute Charm Glitch
The Cute Charm Glitch offers some of the highest shiny odds in the game, with a whopping 21% chance per encounter if the conditions are right. However, it’s extremely difficult to pull off, as the glitch involves manipulation of personality values and Trainer IDs. It’s best to check a professional guide like Smogon’s for a more thorough grasp on how to use the glitch. There are other glitches you can use to manipulate shiny rates -- again, the Smogon article is a great place to get a better understanding of these.
Shiny Charm
The Shiny Charm is one of the most useful items for shiny hunting. Introduced in Black and White 2, it makes it much more likely to find a shiny — plus it stacks with other hunting techniques, like the Masuda Method. In Black and White 2, the odds are roughly triple that of the standard odds, with a 1/2731 of finding a shiny on any given encounter. In later generations, because of the increased base shiny rate, the Shiny Charm now raises odds to about 1/1365. When combined with other methods like the Masuda Method and Dynamax Adventures, the odds are even higher. In Scarlet and Violet, Wild Tera Pokemon and Pokemon found in Tera Raids are not affected by the Shiny Charm.
Note that in Brilliant Diamond and Shining Pearl, the Shiny Charm only works when breeding.
Friend Safari
The Friend Safari is a postgame location in X and Y that has Pokemon not ordinarily obtainable in Kalos. It also offers increased shiny odds, at approximately 1/512 or a 0.19% chance. The Shiny Charm does not increase odds for this method.
Consecutive Fishing
Exclusive to Generation 6 (XY and ORAS), consecutive fishing (commonly called chain fishing) is a mechanic that offers significantly higher shiny odds. If you don’t move from your fishing location, always come up with a bite, and timing your pulls correctly, you can get a streak. The shiny odds cap at around 1/96 or about 1% after twenty consecutive battles, some of the best odds in the series. Thonky’s guide is a great place to learn more about chain fishing.
DexNav
In Omega Ruby and Alpha Sapphire, the DexNav can be used to increase shiny odds. The higher your chain goes, the higher chance there is of the game forcing the next Pokemon to be shiny. This is displayed in the image below from Bulbapedia: