This is my favorite version of the Mew glitch, as it allows you to capture Mew before battling Misty, and, if performed at the recommended time, can yield double results.

The Mew Trick, Cerulean City Version
Notes:
For optimal results, begin this trick after beating Nugget Bridge, but before entering Misty's Gym.
Don't be afraid to reset and try again if you mess up and/or things start going badly for you!
As stated below, this is only confirmed in Pokemon Red Version and Pokemon Blue Version. I do not guarantee success in Yellow Version or the japanese Blue Version. Feel free to try it if you want, though.
This trick does not work for Pokemon FireRed, nor Pokemon LeafGreen. Only the GB/GBC versions; Red and Blue.
There is actually a video of this on youtube. I didn't make it, but it shows that this trick does work on Yellow version.
I didn't invent, nor discover this trick, I just made this guide for it.

Preparations:

Pokemon:
Have at least one Pokemon level 8 to 12. It needs to be at least level 8 so it doesn't get KO'd by Mew, but at or below level 12 so it doesn't KO Mew. This Pokemon could actually be any type you like, as level 7 Mew only knows Pound, but I recommend a Paras, for things like Sleep Powder and Stun Spore. You'll use this Pokemon to weaken Mew. Hitting Mew with paralysis is probably best, as it won't accidentally KO it, nor will it wear off. If you have Bulbasaur, Ivysaur, or Butterfree, they could inflict the status condition as well.
Have at least one Pokemon level 16 to 20. If and when you see the necessity to switch out your Mew-weakening Pokemon for another, you'll want a Pokemon to endure Mew's constant barrage Pounds. (it hits pretty hard for a little floating cat!) I recommend a Geodude for this, as it is the only Rock-type (and thus resistant to Normal) Pokemon you can have at this point without trading.

Items:
Have at least 10 Pokeballs, more if possible, per Mew. Use Great Balls if you can. Mew is a legendary after all, and though (from my experience) it is not as hard to catch as the others, it will still eat up a lot of your Pokeballs.
Have around 5 Potions as well, maybe a Super Potion, too. Mew is a hard hitter, even at level 7, so don't let it wipe you out!

Process:

Step 1: First, get to Cerulean City, beat your Rival there, clear out Nugget Bridge and catch an Abra. DON'T battle the Junior Trainer hiding in the grass left of the Nugget Bridge, the Youngster with a Slowpoke on Route 25, Misty, or the first male Swimmer in her Gym.

Step 2: Visit the Cerulean Pokemon Center and use it, even if you don't need to.

Step 3: Get directly above the aforemetioned Junior Trainer, but so that he's just off the bottom of the screen, save, (to reset if you mess up at any point,) then walk downward one space and pause immediately. Use Abra's Teleport to warp back to the Cerulean Pokemon Center.

Step 4: Go and fight the Slowpoke Youngster, and no one else. Make sure he has to move at least one step to reach you. Use Teleport again.

Step 5: Walk onto Nugget Bridge, and keep heading north until the game involentarily opens the start button menu.

Step 6: Press B to encounter a wild Lv. 7 Mew!

For another Mew:

Step 1: Repeat, but this time, use the first trainer in Misty's Gym (he has Horsea and Shellder, I believe) instead of the Slowpoke Boy. Make sure he has to move at least one step to reach you.

Step 2: Head north across Nugget Bridge again.

Step 3: Press B when the pause menu comes up.

Step 4: Get another Mew!

It is important for you to know that I'm not completely sure about the Gym Trainer part. Also, I have only confirmed this in Red and Blue; I do not guarantee these results in Yellow Version. Not really sure about Green or japanese Blue either. I hope it works for you, though!

Enjoy capturing and owning the elusive psychic feline Mew!

~Surskit of Destiny

I'd be interested to see the video of this.

I have done part one of this method before, and it works. but i've never tried the gym trainer one.

I don't doubt that it's possible, I'd jsut like to see it before I do it. It makes me happy, though, that I can get a Mew in one of my games and know I didn't cheat.....technically...

lol, yes, its a 'legit' way to get a mew, cuz technically gamefreak put it on there on purpose cuz they never make mistakes, do they? *cough cough* yes they do *cough cough*

Stop poking holes in my arguments! XD

Hey they may not have put it there on purpose, but they didn't exactly do the best job on making sure it wasn't there.

yep. but i loved missingno. and the truck. and all the other lil glitches throughout the game. they just dont code like they used to nowadays.... why must they code so well now????? WHY???? *falls to knees, covers face in hands, sobs*

Haha, it makes you feel kind of like a rebel. going and doing these things gamefreak didn't want you to ;D It also makes you feel slightly clever...'oh yes i got me when i shouldnt' Haha i love these glitches, unfortunatly, i can't do the multiplyer cheat..i only have yellow >:

Heh, I saw this glitch performed somewhere before but don't remember where...& yeah, it really does work...

From what I know, It was the creator of Poke'mon who put Mew there, just for fun. No one in Gamefreak knew about it. Only the creator knew about Mew from what i know.

this glitch actually was discovered in 03.

the more publicized version was found later.

You don't know if it works in Yellow? Does anyone else here know if it does then?

Know what, I'll test it myself and tell you all if it works. Would you please point me to the video first though?

I find it interesting that this method is so different from the one in the psydex though. It says you have to have already defeated Misty- as well as Surge- since you'd need Fly for it. Teleporting would SO much easier.

I just tried it on my Yellow version... it DOES work on Yellow!