The Devastation of Splash
Short Version: Magikarp's Splash attack had a 0.001% chance of delivering an insta-kill
Why We Believed It: The promise of seeing a Magikarp destroy your friends' most powerful Pokémon was too awesome to pass up
Ignoring the potential benefit of evolving one of these useless turds, Magikarp is bar-none the most worthless Pokémon one can own. Not only are its stats super pathetic, but its primary move, Splash, literally does nothing at all. In summary, Magikarp is about as helpful in a battle as a plank of wood, except for the fact that in a pinch you could use a plank of wood to hit something over the head.
This general uselessness is probably what spawned the rumor that under certain circumstances, or at incredibly infrequent intervals, Magikarp's Splash attack would instantly kill any Pokémon it was facing. Folks likened it to an even more extreme version of Fissure or Guillotine which made it all the more believable considering that those moves also follow the same logic.
My favorite part of the rumors surrounding the Splash ability was the notion that somehow the Magikarp that the con man in Route 4's Pokémon Center sells the player is the only one who actually has the secret insta-kill power. It just goes to show the mental gymnastics people are willing to go through to avoid realizing they've been played for a fool.