Attack	Pokémon
Except for Surfing Pikachu, you will recieve a new Pokémon with the given ability. None of these moves are breedable. Psyduck, Gligar and Farfetch'd will have fixed ID #'s and Stadium as OT.
Surf	In the original Pokémon Stadium, a Pikachu can be given the Surf ability. Take a Pikachu (not Raichu) through the R2 Master Ball circuit without using a Continue.
Dizzy Punch	In Crystal, the old man in the Day Care will give you an Odd Egg. It will hatch into one of the babies; Pichu, Cleffa, Igglybuff, Smoochum, Elekid, Magby, Tyrogue, or Togepi, with Dizzy Punch. In the Japanese version, there is a 50% chance of this Pokémon being Shiny, or having DVs of all 0. In the American version, the chances of getting the Shiny are MUCH lower.
Amnesia	When you have used all 151 Pokémon in the Stadium, Gym Leader Castle or Mewtwo modes, you will recieve a Psyduck with the Amnesia ability.
Extremespeed	After recieving the Risingbadge in Crystal, the Dragon elder will give you this Dratini. It will not have Extremespeed if you answered any questions incorrectly.
Earthquake	Defeat your rival in Stadium GS in R2 mode to win a Gligar with Earthquake, as well as Poison Sting, Wing Attack and Counter.
Baton Pass	Defeat your rival in Stadium GS in R1 mode to win a Farfetch'd with Baton Pass, as well as Slash, Swords Dance and Agility.
The following list of Pokémon can be taught these abilities by a man in the Game Corner in Pokémon Crystal. It is unknown if these abilities are breedable.
Flamethrower	Charizard, Nidoqueen, Nidoking, Clefable, Ninetales, Wigglytuff, Arcanine, Machamp, Golem, Rapidash, Slowbro, Muk, Marowak, Lickitung, Weezing, Rhydon, Kangaskhan, Magmar, Tauros, Gyarados, Flareon, Aerodactyl, Snorlax, Moltres, Dragonite, Mewtwo, Mew, Typhlosion, Togetic, Slowking, Dunsparce, Magcargo, Octillery, Houndoom, Blissey, Entei, Tyranitar, Houou
Thunderbolt	Raticate, Raichu, Nidoqueen, Nidoking, Clefable, Wigglytuff, Persian, Primeape, Magneton, Muk, Gengar, Electrode, Lickitung, Weezing, Rhydon, Kangaskhan, Starmie, Mr. Mime, Electabuzz, Tauros, Gyarados, Lapras, Jolteon, Snorlax, Zapdos, Dragonite, Mewtwo, Mew, Lanturn, Ampharos, Aipom, Misdreavus, Girafarig, Dunsparce, Snubbull, Porygon2, Elekid, Miltank, Blissey, Raikou, Tyranitar, Lugia, Houou
Ice Beam	Blastoise, Raticate, Nidoqueen, Nidoking, Clefable, Wigglytuff, Golduck, Poliwrath, Tentacruel, Slowbro, Dewgong, Cloyster, Kingler, Marowak, Lickitung, Rhydon, Kangaskhan, Seaking, Starmie, Jynx, Tauros, Gyarados, Lapras, Vaporeon, Omaster, Kabutops, Snorlax, Articuno, Dragonite, Mewtwo, Mew, Feraligatr, Lanturn, Azumarill, Politoed, Quagsire, Slowking, Qwilfish, Sneasel, Piloswine, Corsola, Octillery, Delibird, Mantine, Kingdra, Porygon2, Miltank, Blissey, Suicune, Tyranitar, Lugia