Mew:
on red/blue
When you're in vermilion city for the first trade a pokémon that can learn surf/ strength. use
surf on the deck, there use strength against a truck: mew should appear:FAKE
This truck does exist: first make sure the SS. Anne is gone, go stand diagonal from the man guarding the deck move right and press start when moving. save and turn off the power. Turn on the power and use surf. You should now surf on the man, surf trough and you're on the deck. You can re-board the SS. Anne and see the truck if you use surf.
This same truck is believed to be on leaf green/fire red but I have no proof
(Not real)

Celebi:
this only works on the Japanese version of Cristal
somewhere in the game a girl tells you that as you collect all the badges you can get a present: this is the GS ball give it to Kurt, the next day the wind starts to blow real hard and Kurt gives you back the GS ball: place it in the shrine in elix forest and celebi will appear. this event is not erased on the English version, just the GS ball is, so if you have a gameshark you can get the GS ball and catch celebi

missingno:
on red/blue
My personal favourite, if you talk to the coffee man in veridian city and say no, watch his battle, then go to the shore at the seaform islands and surf up and down, missingno, M or pokémons from more then lv100 will appear, if you have fought missingno, the sixth item in your item pocket will be there for over a hundred times, good for copying master balls.
'M is almost always on lv.0, he evolves into a kangaskan on lv.1, Because 'M can learn almost every attack, you can get kangaskan with fly or something like that. the fun thing about 'M is that if you catch him he'll be in your party but the battle hasn't ended, if you catch him again you'll catch a ditto instead and the battle will finally end
To get the other missingno's name yourself something with a w, x, or y in the 3rd, 5th, and 7th letter positions. "w" lets you get Kabutops skeleton, "x" lets you get Aerodactyl skeleton, and "y" lets you get the Ghost. This won't work if you use capital letters
if you catch missingno(#0) and carry him in your party, all your pokémon will be turned around and will look strange if you use them in battle, you can also lose all your pokémon in your boxes, your Hall of fame will be messed up: so don't catch him!!!
Extra: he looks like a substitute character and you can't use him in battles on pokémon stadium and he'll turn into a ditto with transform only on pokémon stadium 2
on yellow
trade missingno to your yellow to be able to catch missingno there too (the glitch was supposed to be erased)
on ruby/sapphire
he is believed to be in the wild but I can't prove this, he's would be called ????? and be like a giant question mark (a cheat device would be needed for this) he looks like a golden bulbasaur in you party
missingno or 'Mthe othersthe new onemissingno when he's in your party

glitch city:
on red/blue
to go to a strange glitch city do this: go to the safari zone, enter and turn directly around, then say now to the guy the guy that asks or you want to leave, then when you're back in save and turn off the power. put back on and leave, the guy will ask you or you want to enter the safari zone, say no. go out of the house and walk around, after 500 steps the PA:ding-dong will appear, you'll be back in the safari house, go out and you'll be in glitch city. to exit: use fly
If the PA goes off on the bicycle road, you'll be in a glitch city without your bike but you'll keep on walking down, like in the bicycle road

Colour glitch:
on gold/silver
Talk to the machop in veridian city, after that, don't move, push start and check the coin  case, your game will restart (like putting the power on, this will not erase your game) but it will be totally discoloured, it is fun to look at, end it by putting off the power. If you try this on pokémon stadium2 you're game will freeze or it will give a message that you're game has a fatal error and that you have to start a new game, but your game will still work, there are no real dangers for your game

cloning
on red/blue/gold/silver
trade the pokémon you want to clone and a pokémon that you won't mind losing, at the end of when the game is saving turn off the power of the game that originally had the pokémon you wanted to clone, you'll lose the other pokémon but your pokémon will be on both games
on gold/silver
Deposit the pokémon you want to clone in a box other then box1, then change the box to box1, when the game is saving and you mustn't turn of the power, turn of the power, the pokémon will be in your box and in your party
on crystal
the same as  on gold/silver but you have to have 5 pokémon in your party before doing this
on emerald
1. Deposit the pokemon you want to duplicate in a box at any PC, and any box.
2. Save.
3. Take out the pokemon you want to duplicate from the box.
4. Go to the Battle Tower and talk to the lady on the far right, the one who hosts the Wireless battles.
5. Challenge the event and pick two eligible pokemon. It does not matter which event it is, as long as you have eligible pokemon.
6. The lady will ask if you want to enter those two pokemon. Select yes, and a two-second pause will occur.
7. When it asks if you want to save, say no and reset.
this should be harmless, but do not turn off the power in that 2 second pauze

walk on the rocks:
on ruby/sapphire
go to sootopolis and stand in front of the the stairs underneath the top stairs, walk on them and walk left, you're now walking on the rocks =D

surf on statue:
on red/blue
enter the pokémon league, you'll begin in Lorelei's room, there you'll find a statue on the left side but not on the right side, you can surf on the statue

others:
A one in a lifetime glitch: on my red I once traded my Mew for a butterfree, then I played on my red with my Pokémon stadium. I fought a drowzee, I've beaten him and My butterfree grow from lv.28 to lv.100. Strange!
Another one. On emerald my jirachi once fainted without losing HP: he's been hit by an earthquake. without that his HP lowered he just fainted.

How???:
pokémon had a release date but the makers had trouble's with the death line, on red & blue you'll still find mistakes and errors, if you have a gameshark, you can even discover the not fully removed errors, like there used to be a surfboard that would enable you to surf without the HM, you could buy it in every store, but it was removed at the last moment. if you use the gameshark you could get a nidoking holding this item (or so I heard)