One-Offs Guide

Introduction

First of all, what are One-Offs? They are pokémon, most of them Legendaries, that you only have one chance in the whole game to catch. They are either sitting in a place waiting for you to got to them or running through the map.
Here I will go through each one of them (except starters and 4th Generation ones, since I've never played any of those games so I wouldn't be a reliable source), the places they're in (if they are not running through the map) and the best ways to catch them. I hope this helps you in any way ;D

First Generation

In the first generation there are seven non-Lendaries and five Legendary One-Offs, all of them still:
Quick Access
Hitmonlee | Hitmonchan | Mr. Mime | Eevee | Omanyte | Kabuto | Aerodactyl
Articuno | Zapdos | Moltres | Mewtwo | Mew

Hitmonlee - Fighting

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Where to Find Red/Blue/Yellow and Fire Red/Leaf Green: It's quite simple, what you have to do is beat the Fighting Dojo, leader of the Gym in Saffron next to Sabrina's. You get the change to pick one of two Pokéballs: Hitmonlee or Hitmonchan. Just pick Hitmonlee.

Other Games: Trade from Red/Blue/Yellow/Fire Red/Leaf Green/XD

Hitmonchan - Fighting

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Where to Find Red/Blue/Yellow and Fire Red/Leaf Green: It's quite simple, what you have to do is beat the Fighting Dojo, leader of the Gym in Saffron next to Sabrina's. You get the change to pick one of two Pokéballs: Hitmonlee or Hitmonchan. Just pick Hitmonchan.

Other Games: Trade from Red/Blue/Yellow/Fire Red/Leaf Green/XD

Mr. Mime - Psychic

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Where to Find Red/Blue/Yellow and Fire Red/Leaf Green: You must have an Abra, if you don't go back catch one >3. Now, on your way to Viridian Forest, from Pallet Town, there are some trees that can be cut. Use Cut on them and go to a house in Route 3 where you will be able to trade your Abra for a Mr. Mime.

Other Games: Trade from Red/Blue/Yellow/Fire Red/Leaf Green/XD

Eevee - Normal

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Where to Find Red/Blue/Yellow and Fire Red/Leaf Green: Go to Celadon and enter the Celadon Mansion. From there, go to the furtherest floor up and enter a room with a man. talk to him, get Eevee, and enjoy ^^.

Other Games: Trade from Red/Blue/Yellow/Fire Red/Leaf Green/XD

Omanyte - Rock/Water

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Where to Find Red/Blue/Yellow and Fire Red/Leaf Green: Inside of Mount Moon, you will encounter a Super Nerd. Beat him and you get to pick one of two Fossils. Just pick the Helix Fossil.

Other Games: Trade from Red/Blue/Yellow/Fire Red/Leaf Green/XD

Kabuto - Rock/Water

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Where to Find Red/Blue/Yellow and Fire Red/Leaf Green: Inside of Mount Moon, you will encounter a Super Nerd. Beat him and you get to pick one of two Fossils. Just pick the Dome Fossil.

Other Games: Trade from Red/Blue/Yellow/Fire Red/Leaf Green/XD

Aerodactyl - Rock/Flying

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Where to Find Red/Blue/Yellow and Fire Red/Leaf Green: In Fuschia City, get the Old Amber and then fly to Cinnabar Island and get in the Pokémon Lab. Get the Old Amber to the scientist in the third room and he'll revive it into an Aerodactyl.

Other Games: Trade from Red/Blue/Yellow/Fire Red/Leaf Green

Articuno - Ice/Flying

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Where to Find Red/Blue/Yellow and Fire Red/Leaf Green: Articuno can be found at the cave in Seafoam Islands. From Fuchsia City's Pok?on Center, go south and Surf once you see water. Keep swimming Southwest of that shore until you find an island which should be Seafoam Islands. Go into the cave, push the two boulders into the water, using the little holes, and the boulders will disappear. After you clear the boulders out of your way, swim in the water and you'll find Articuno on a little extension of land.

Other Games: Trade from Red/Blue/Yellow/Fire Red/Leaf Green/XD


Strategy to catch Red/Blue/Yellow and Fire Red/Leaf Green: Articuno is an Ice and Flying pokémon. Rock is 4x strong against it so using a high level Tyranitar, Magcargo or Camerupt will do just fine against it but make sure they have at least one Rock type move like Rock Throw. Nothing too powerful, though, as you may faint it with a couple of Rock Slides. If you don't have such a pokémon, you can opt by focusing on the Flying type and use an Electric pokémon such as Electabuzz or Ampharos. Never fall in the mistake of using a Dragon/Grass type pokémon because Articuno will tear it apart.
So get yourself one of the above mentioned pokémon and try causing it a status problem like Sleep or Burn. If you can, use the attack False Swipe since it's a move that doesn't kill the opponent of the pokémon using it, but may leave it to 1HP... I would count much on it, though, just use it if Articuno has very low HP and you think you'll make it faint if you use any other attack.

Zapdos - Electric/Flying

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Where to Find Red/Blue/Yellow and Fire Red/Leaf Green: You can find Zapdos inside the Power Plant, which is located Northwest of Cerulean City. Once in it, go to the top left corner room and Zapdos should be there waiting for you.

Other Games: Trade from Red/Blue/Yellow/Fire Red/Leaf Green/XD


Strategy to catch Red/Blue/Yellow and Fire Red/Leaf Green: Zapdos is the hardest Legendary bird in matters of type influences: since Zapdos is already an Electric type, leave your Electabuzz/Ampharos/Electrode in the box. You can pack your team with a strong Dewgong or Blissey but make sure they have at least an Aurora Beam or an Ice Beam. Another good strategy is to use a Rock type pokémon since Rock moves are Super Effective against Flying. Again, False Swipe is a good move to use when you feel another attack will make Zapdos faint and try to cause it a status problem like Sleep/Freeze.

Moltres - Fire/Flying

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Where to Find Red/Blue/Yellow: Moltres... probably the most hard to find Legendary bird. You can find it in Victory Road but you'll have to move several boulders on some buttons and go up and down rows and rows of stairs to find its exact location. Once you find yourself in that level of the cave, Moltres is in the center of the division.

Fire Red/Leaf Green: Moltres lies within Mount Ember in these versions.

Other Games: Trade from Red/Blue/Yellow/Fire Red/Leaf Green/XD


Strategy to catch Red/Blue/Yellow and Fire Red/Leaf Green: Well, the obvious weakness here is Rock, since both Fire and Flying are 2x weak to it. So any Tyranitar or if you want to take even more advantage a regular Kabutops/Omastar/Relicanth will do just fine since both their types are resistant to Fire. If you can, use the attack False Swipe if Moltres has very low HP and you think you'll make it faint if you use any other attack. Causing it Sleep wouldn't be a bad idea.

Mewtwo - Psychic

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Where to Find Red/Blue/Yellow and Fire Red/Leaf Green: From Cerulean, head North through the bridge and then go West, towards a small chunk of grass. Get to the edge of that grass place and surf Southwest until you find a cave. Mewtwo lies in that cave, waiting for a brave enough trainer to challenge it. Go inside the cave and find the Legendary pokémon on a small island in the second floor.

Other Games: Trade from Red/Blue/Yellow/Fire Red/Leaf Green/XD


Strategy to catch Red/Blue/Yellow: Well, you must be quite prepared to face a level 70 Mewtwo, this thing is practically unstoppable. Since Red, Blue and Yellow have no Dark type pokémon your advantages are narrowed to Bug and and Ghost types. So pack your team with a powerful Gengar with at least a Night Shade or a move like it. Drop the False Swipe this time, focus more on not getting yourself killed rather than leaving it to 1HP. Try a Hypnosis on Mewtwo, weaken it the most you can and keep throwing Ultra Balls until it wakes up again and repeat the process until you catch it is the strategy that should work best. A Chansey would also be a good choice once you have it at the HP you want; It can hold a couple of attacks and can freeze it.

Fire Red/Leaf Green: The basic guidelines are set in the R/B/Y strategy above, but instead of a Gengar you can use a strong enough Tyranitar or Houndoom and you should do better. A Sneasel or an Umbreon will serve you right, too.

Mew - Psychic

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Where to Find Red/Blue/Yellow: To get Mew you must go to a Nintendo event and obtain it via a wireless device, ooooor... you can use the Mew Glitch.

Emerald: To get Mew you must go to a Nintendo event and obtain the Old Sea Chart through a wireless device. You must then access World's Edge Island with the Old Sea Chart and get in the forest. Mew will run from you once he sees you enter the forest so what you have to do is chase him.

Other Games: Trade from Red/Blue/Yellow/Emerald
Strategy to catch Emerald: Use the same strategy as the Mewtwo one but be careful because Mew is 40 level weaker than Mewtwo.

Second Generation

The second generation shelters a few less one-offs but more legendaries than 1st Gen.
Quick Access
Sudowoodo
Raikou | Entei | Suicune | Lugia | Ho-Oh | Celebi

Sudowoodo - Rock

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Where to Find Gold/Silver/Crystal: Blocking your passage through Route 36, there'll be a small tree. You must then use the key item SquirtBottle in order for it to jump and attack you.

Emerald: South-West at the Battle Frontier, go near the Battle Palace. From there cross the river and you'll see a dejà vú from Gold/Silver/Crystal, a tree blocking your passage. Go up to it and and use the Wailmer Pail to sprinkle some water on the tree; that'll cause Sudowoodo to attack you.

Other Games: Trade from Gold/Silver/Crystal/Emerald/Colosseum

Strategy to catch Gold/Silver/Crystal/Emerald: If you have a fairly strong Water or Grass type the battle should be easy, and seeing Sudowoodo's Speed is nothing special really, take advantage of that to put it to sleep or paralyze/freeze it in the first turn if you can. Once weakened, a couple of Great Balls should do the job perfectly.

Raikou - Electric

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Where to Find Gold/Silver/Crystal: First of all, you must go to Burned Tower in Ekruteak and release the 3 Beasts; Raikou will then appear randomly through the grasses of Johto.

Fire Red/Leaf Green: Raikou will only appear to you if you pick Squirtle as starter.

Other Games: Trade from Gold/Silver/Crystal/Fire Red/Leaf Green.


Strategy to catch Gold/Silver/Crystal/Fire Red/Leaf Green: Seeing that Raikou is a runner, you'd better have a fast pokémon yourself. Because there are no abilities in G/S/C, you can't, for example, Shadow Tag it and prevent it to flee from the battle so you'll have to run into it several times and keep inflicting damage until it's weak enough to throw an Ultra Ball. A high-leveled pokémon like Electrode or Gengar would be ideal due to their Speed stat.

Entei - Fire

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Where to Find Gold/Silver/Crystal: First of all, you must go to Burned Tower in Ekruteak and release the 3 Beasts; Entei will then appear randomly through the grasses of Johto.

Fire Red/Leaf Green: Entei will only appear to you if you pick Bulbasaur as starter.

Other Games: Trade from Gold/Silver/Crystal/Fire Red/Leaf Green.


Strategy to catch Gold/Silver/Crystal/Fire Red/Leaf Green: Seeing that Entei is a runner, you'd better have a fast pokémon yourself. Because there are no abilities in G/S/C, you can't, for example, Shadow Tag it and prevent it to flee from the battle so you'll have to run into it several times and keep inflicting damage until it's weak enough to throw an Ultra Ball. A high-leveled pokémon like Electrode or Gengar would be ideal due to their Speed stat.

Suicune - Water

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Where to Find Gold/Silver: First of all, you must go to Burned Tower in Ekruteak and release the 3 Beasts; Suicune will then appear randomly through the grasses of Johto.

Crystal: First of all, you must go to Burned Tower in Ekruteak and release the 3 Beasts; Suicune will then be released and there are 4 places where you can meet it, but only one where you can catch it: meet it near the National Park, in Mount Mortar and in Cianwood. Suicune will then await you in the Tin Tower.

Fire Red/Leaf Green: Suicune will only appear to you if you pick Charmander as starter.

Other Games: Trade from Gold/Silver/Crystal/Fire Red/Leaf Green.


Strategy to catch Gold/Silver/Crystal/Fire Red/Leaf Green: I reccomend a strong and fast pokémon with Grass/Electric attacks and False Swipes, perhaps a Zangoose with Thunderbolt, False Swipes, Giga Drain and Slash/Crush Claw or a pure Electric type like Electrode/Raichu if you're willing to face it without False Swipes.

Lugia - Psychic/Flying

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Where to Find Gold/Silver/Crystal: The creature can be found resting in the Whirl Islands after you've obtain the Silver Wing.

Fire Red/Leaf Green/Emerald: You'll have to reach Navel Rock (requires Mystic Ticket) to catch it.

Other Games: Trade from Gold/Silver/Crystal/Fire Red/Leaf Green/Emerald.


Strategy to catch Gold/Silver/Crystal/Fire Red/Leaf Green/Emerald: This would be easy if Lugia didn't have the move Rest because it has lot's of weaknesses, probably an 70+ Gengar with Thunderbolt, Hypnosis, Nightmare, Shadow Ball/Dark Pulse.

Ho-Oh - Fire/Flying

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Where to Find Gold/Silver/Crystal: The creature can be found resting in the top of the Tin Tower after you've obtain the Rainbow Wing and caught all the legendary dogs.

Fire Red/Leaf Green/Emerald: You'll have to reach Navel Rock (requires Mystic Ticket) to catch it.

Other Games: Trade from Gold/Silver/Crystal/Fire Red/Leaf Green/Emerald.


Strategy to catch Gold/Silver/Crystal/Fire Red/Leaf Green/Emerald: Ho-Oh is one of the easiest legendary one-offs you'll catch: A fairly strong Poliwrath/Kabutops with Surf, Rock Slide, Rest and Aqua Jet will do the trick.

Celebi - Psychic/Grass

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Where to Find All Games: Obtain from a Nintendo Event