In 'WALL-E', what is the name of the spaceship?

In WALL-E, the spaceship is named the Axiom.

The Axiom is a massive starliner operated by the Buy n Large corporation. It carries humanity away from an uninhabitable Earth after pollution and consumer waste make the planet unsafe. The film's story takes place in the year 2805, roughly seven centuries after humans left Earth, and the ship has become a fully automated home for generations of passengers.

WALL-E reaches the Axiom by clinging to the rocket that collects EVE after she discovers a living plant. Once aboard, he encounters the ship's captain, B. McCrea, and the autopilot computer AUTO. The Axiom's comfortable systems have made its passengers physically dependent and disconnected from their surroundings, making the ship an important part of the film's commentary on consumerism and technology.

Discovery is a common mix-up because it is the name of the spacecraft in 2001: A Space Odyssey. BnL is also not the ship's name; it is the abbreviation for Buy n Large, the corporation that owns and operates the Axiom. The vessel is sometimes described as a starliner or mothership, but Axiom is its proper name.