Unfortunately, it is impossible to catch a mew on Pokémon Silver. You can, however, obtain a Mew through either using a GameShark cartridge cheat (code 0197EDD0) or by trading from Pokémon Red/Blue after using the Mew glitch.
- Restart your game on Pokémon Red, Blue or Yellow
It is absolutely vital that you restart the game. This is because you may have already done things that would prevent you from being able to get a Mew, such as defeated certain trainers, if you use a previous save.
- Progress through the game until you get HM02 - Fly
HM02 Fly can be obtained by visiting the small house west of Celadon City, close to where Snorlax is enjoying a nice kip. Once there, speak to the girl inside, and she will give you the HM.
- Avoid fighting certain trainers
It is incredibly important that you do not fight the following trainers: The 4th trainer after crossing the Nugget Bridge on the way to Bill's house (Route 25) and the gambler between Lavender Town and Saffron City.
- Equip a Pokémon with fly, and head to the underground pass
As long as you have a Pokémon with fly in your party, head to the underground pass on Route 8, and just in front of the door, save your game. Saving after this point will disrupt the process and prevent you from obtaining a Mew.
- Glitch the gambler
Walk down one step, and press the start button. If you have done this correctly, the gambler will not have seen you. Select the Pokémon that knows fly, and travel to Cerulean City. Just as you leave, the gambler will notice you, but at this stage, it is too late for him to pick a fight. You may realize that your start button is disabled once you reach the city. This is entirely normal and actually vital to the glitch.
- Beat the youngster's Slowpoke
Head across the Nugget Bridge, and battle the 4th trainer on the way to Bill's House. He will have a Slowpoke, which should be pretty easy to defeat. Once you have done this, the start button will work again. Fly to Lavender Town, and head to Route 8.
- Find the mysterious pop-up menu
As you enter the narrow path from Lavender Town, the menu will pop-up by its own accord. This is supposed to happen, so simply press B and close the menu. As soon as you do, a Level 7 Mew should pop up, and you start throwing Pokéballs galore. After you have caught the Mew, make sure to save your game and head to the nearest Pokémon Center.
- Trade from Pokemon Red/Blue to Pokemon Silver
For this, you will need two Gameboys, or a friend willing to help, and a link cable. All that is left to do is trade the newly caught Mew onto your Pokémon Silver cartridge, ready to train up into the ultimate fighting machine.