Shiny Pokémon

Shiny Pokémon are Pokémon with different coloration than the normal versions of the Pokémon although they have no stat differences at all. Shiny Pokémon have been included since Generation II in Gold and Silver in which the first shiny Pokémon that was introduced was a Red Gyarados. Shiny Pokémon are considered very rare. The term "shiny" is a reference to their difference in color and their sparkling animation and sound effect when they enter into battle. The term was unofficial prior to Generation V, when the Pokédex began cataloging shiny Pokémon and using the term itself.

Encountering
In Generation II, a Pokémon randomly assigned 5 integers between 0 and 15 to determine their base stats. (HP, Attack, Defense, Special, and Speed). If all 5 numbers are equal to 10, the Pokémon's color scheme will be set to 0. The chance of seeing a shiny Pokémon is 1 in every 8192, or a probability of 0.01220703% during each encounter. From Generation III onwards, shiny Pokémon are determined by other factors such as the Trainer ID number and the personality value of the Pokémon. Despite this change, the odds of randomly encountering a shiny Pokémon is still 1 in 8192. As of Pokémon Black 2 and Pokémon White 2, the Shiny Charm can be obtained to significantly increase chances of encountering a shiny Pokémon. This stacks with most methods listed below.

Methods
There are a few different methods to increase the chances of obtaining a shiny Pokémon.

Chaining (Gen IV)
This method uses the Pokéradar in Diamond, Pearl, Platinum, X and Y to encounter chains of the same Pokémon. Here's a few tips (Non-Gen VI):
 * If the bush just shakes, it is a Sinnoh Pokémon.
 * If the bush has a whitish shake, it might be a non-native Pokémon.
 * Never use it in water, caves, or tall grass.
 * The bush with the same type of shake as the first Pokémon one has battled that is the farthest away within a four by four grid is most likely the same Pokémon.
 * The likelihood of finding a shiny Pokémon increases as the chain increases, maxing out at 40. Pulsing, sparkling grass indicates a Shiny Pokémon is in that bush.

For the Gen VI games, specifically ORAS, chaining has been made much easier thanks to the DexNav application on the PokéNav+. Using the DexNav function, players can see which Pokémon is in a patch of grass without having to risk encountering it. Likewise, scaring off a Pokémon using the DexNav (i.e. failing to sneak up on it) does not break the chain.

Masuda Method (Gen IV)
To use this method one must breed two Pokémon from games in different languages. Using this method improves the odds of hatching a shiny Pokémon from 1/8192 to 1/2048 or 1/1365.3, in Generation IV or Generation V, respectively. This method was the brainchild of Junichi Masuda, director of GameFreak.

Soft Reset (Gen. II to Gen. VII)
One method which is easy enough (but often very time consuming) works for starter Pokémon and most Legendary Pokémon. Basically, the method is to stand in front of the legendary Pokémon the player character will catch or the starter Pokémon they will take and save. If one doesn't get a shiny Pokémon when getting the starter Pokémon or battle the legendary Pokémon, soft reset the game by pressing A+B+Start+Select on the GBA, L+R+Start+Select on the DS and L+R+Start on the 3DS. This method can often require over 1000 resets before getting a shiny Pokémon.

In-Game Shiny Pokémon
In-game shiny Pokémon are met as part of the main story in some Pokémon games. For example, the Red Gyarados appears in Gold, Silver, Crystal, HeartGold and SoulSilver, as part of a story arc, and can be caught in a way similar to most legendary Pokémon.

In FireRed and LeafGreen, a Trainer with a Shiny Espeon can be battled inside the Trainer Tower on one of the Sevii Islands. There is also another trainer with a shiny Meowth.

In Black and White 2, the player can catch a Shiny Haxorus in the post-story. The player can also receive a Shiny Gible in Black 2 and a Shiny Dratini in White 2.

In Sun and Moon, after becoming the Champion, a man in the Seafolk Village Pokémon Center will battle the player character with his shiny Exeggcute.

Anime

 * Jigglypuff: It was in fact a shiny version of a Jigglypuff in reference to the colors of the shiny version of this Pokémon.
 * Noctowl: Ash Ketchum caught one while on his travels through Johto. It also seems to be about half the size of other Noctowls.
 * Shuckle: While in Johto, Ash and co. discovered a Shiny Shuckle.
 * Gyarados: First seen in the Lake of Rage and was caught by Lance. It was also seen again during the battle between Kyogre and Groudon. Lysandre also owns one but it died with him when he was killed by Zygarde's Core Enforcer.
 * Magneton: Owned by Pokémon Trainer Jackson when he battles Ash in the Silver Conference.
 * Swellow: Winona, the Fortree City Gym Leader has a Shiny Swellow.
 * Donphan: A Shiny Donphan has been shown in the Hoenn Region on Dontoe Island.
 * Magikarp: Was seen briefly in the episode "Judgement Day" and "Going for the Gold!".
 * Dustox: Jessie's Dustox fell in love with a male Shiny Dustox and then was released similar to how Ash released his Butterfree. (Except Jessie smashed Dustox's Poké Ball so it could go with its true love)
 * Shiny Metagross: In the episode "Noodles! Roamin' Off", James in Team Rocket went out to see this strangely colored Metagross that people had been talking about, and was promptly attacked by the shiny Metagross. Another Shiny Metagross is owned by the Hoenn Elite Champion, Steven.
 * Pichu: A Shiny Pichu appeared in the fourth ending of the Diamond and Pearl series: Get Fired Up, Spiky-eared Pichu!.
 * Ditto: A shiny Ditto appeared in "Dealing With Fierce Double Ditto Drama" under Narissa 's ownership.
 * Ariados: In the Special Episode "Dawn's New Journey", Dawn and Cinco encountered a swarm of Ariados as well as a shiny one using Psychic so the others could wrap Dawn and the others with String Shot.
 * Entei, Raikou, and Suicune: the three legendary beasts were shiny guardians of the city in the Pokémon Movie: Zoroark: Master of Illusions
 * Hydreigon: Owned by Carlita in White, Victini, and Reshiram. It is used in the tournament In that movie.
 * Golurk: Owned by Carlita in White: Victini and Zekrom. It was used to guard Juanita's trailer that contained Victini dolls and candy.
 * Onix: The Pokémon was seen near the end in, "Expedition to Onix Island" after Meloetta uses her song to calm down the other attacking Onix. It seemed to have been the leader of the Onix.
 * Red Genesect: This Shiny Legendary Pokémon is the leader of the Genesect Army in, MS016: Pokémon The Movie - Genesect and the Legend Awakened.
 * Hawlucha: Owned by Carl in "When Light and Dark Collide!"
 * Rayquaza: A Shiny Legendary Pokémon that can Mega Evolve in MS018: Pokémon The Movie - Hoopa and the Clash of Ages.
 * Dragonair: Owned by Amelia.
 * Phantump: Leader of a group of wild Phantump in XY117.
 * Pikachu: This Shiny Pikachu appeared at the end of XY117 during the Poké TV segment that focused on Shiny Pokémon.
 * Gengar: Owned by Jarvis in MS019.
 * Gardevoir: Owned by Chymia in MS019.
 * Charjabug: Owned by Hayate

Trivia

 * In Generation II games shiny Pokémon can be slightly stronger than their normal colored counterparts. However, this is not true for later generations.
 * There are examples of Pokémon which, although colored differently, are not officially shiny, such as gender or form differences in regular coloring such as in Hippowdon, Jellicent, and Gastrodon. However, these are not shiny variants, but each gender or form will have its own shiny color that differs from the usual color for their gender or form.
 * The first alternately colored Pokémon to appear in the anime series was seen by Ash in the first season when he released his Butterfree so it could find a mate.
 * In Pokémon Mystery Dungeon: Red Rescue Team and Blue Rescue Team, two Kecleon own a shop in the town square. One is a dark purple color.
 * The Pokémon which reside in parts of the Orange Islands are colored differently to other Pokémon. The color changes are due to climate and natural adaption according to Professor Ivy.
 * In the Anime, "In the Pink" was just that apparently, eating the berries there turned Pokémon pink. However, if the Pokémon stopped eating the berries, they would discontinue being pink. Ash's Pikachu was temporarily pink due to this.
 * The term "Shiny Pokémon" was originally a fan name, however, it became so popular that it became the official name for differently colored Pokémon.