The Falcon Flyer is Captain Falcon's spacecraft used for finding crime activity. It comes with a computer as well. It is also used for carrying the Blue Falcon. It first appeared in the F-Zero manual but made its first appearence in-game in Super Smash Bros. Melee on the Big Blue stage. It also appears in Super Smash Bros. Brawl during the Subspace Emissary, in the boss battle with Meta Ridley (from the Metroid series).
Super Smash Bros. Brawl
In the Subspace Emissary, the falcon flyer appears when Captain Falcon snaps his fingers. The ship is used to escape from the factory that is about to blow. During the scape the ship is attacked by Meta Ridley. After this the ship meets with the Halberd at the desert.
It's next appearance is when various ships from several Nintendo series are trying to stop the villains. After it enters the Subspace is unknown what happened to the ship. However, it appears everyone inside it jumped out of the ship.
First apparition in the F-Zero manual