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
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.
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.
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!
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
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