Corrected entry: For this glitch, advance through the game until you get the FLY HM and get a Pokemon that can learn how to fly. Also, do not fight the trainer in the Cape that only has a Slowpoke, nor the trainer that is looking up outside the Underground Path (right entrance). Go to this entrance of the Underground Path and face down. Press down and immediately press Start. You'll be in the trainer's sighting, but he won't react to this. Go to Pokemon, choose the Pokemon that knows how to Fly and fly to Cerulean City. The trainer will now notice you, but you'll fly away. In Cerulean City, the Start menu won't work. Now go to the Cape and let the Slowpoke trainer see you and walk to you. Defeat his Slowpoke and the Start menu will work again. Fly to Lavender town, and go to the route that heads to the left. When you enter the route, the Start button will pop up for no reason. Press B, and a wild Pokemon animation will show up. From nowhere, a Mew, an impossible-to-get Pokemon, will want to fight with you.
Corrected entry: A trainer near the Nugget Bridge says: I saw your "Feat" in the grass. When it should say "feet". This is also present in the Fire Red and Leaf Green Versions.
Correction: The Trainer means "feat" as in an achievement, referring to the player having defeated the Nugget bridge trainers.
Corrected entry: The first question of the Cinnabar GYM is "Does Caterpie evolves into Butterfree?". According to the GYM the answer is YES. However, Caterpie doesn't evolve into Butterfree. Caterpie evolves into Metapod and then Metapod evolves into Butterfree, later. So saying that Caterpie evolves directly into Butterfree is a mistake.
Correction: This doesn't work.
Yes, it does. It's called the Mew glitch, and it's a very known glitch. Instructions here: http://pokedex.kary.ca/catchmew.shtml.