I'm sorry if this is common knowledge, but I thought it would be good to have a thread about it.

Before I post it, just know that this glitch 100% works.

SAVE BEFORE STARTING THIS IN CASE YOU SCREW UP.

Okay, here's what you do...

Play the game until you can fly. IMPORTANT! YOU MUST SKIP THESE TRAINERS, AS IN DO NOT FIGHT THEM...
- The youngster with a lv. 17 Slowpoke on Route 25 (North of Cerulean) He is the 4th trainer after you cross Nugget Bridge.
-The gambler with 2 Poliwags between Lavender and Saffron (Route 8). He is standing just east of the entrance to Saffron.

Here what you need to perform this glitch...

Pokemon that knows FLY
A level 5 or 6 pokemon (to weaken Mew
A pokemon that can put Mew to sleep
A load of balls (pokeballs that is. Great Balls work the best Very Happy)

On Route 8, stand in front of the door to the underground path. Walk down one step (in front of the gambler) and push start IMMEDIATELY. Then select to fly to Cerulean City. If performed correctly, you will turn into the bird, the gambler will see you (getting a ! above his head) Before he walks up to you, you'll fly away.

Once in Cerulean City, you'll notice that you're START button won't work. Don't fret. Walk up to the Youngster who you should've skipped and battle him. (IF YOU WALK RIGHT UP TO THE YOUNGSTER, YOU'RE GAME WILL FREEZE. GET FAR ENOUGH FROM HIM SO THAT HE HAS TO WALK TO YOU.)
After defeating the youngster, you're START button will work again. Fly to Lavender and head west. As you pass the narrow passageway due west of Lavender, the START menu will pop up by itself. Press B to exit it, and a Lv. 7 Mew will appear. All it knows is pound. Catch'em Very Happy After catching Mew, you're game will work normally again.

Purple Summer has confirmed that this does work in Yellow, along with the Red and Blue. Have fun!

~Eon