The Bucket would first appear in Mario & Wario. They would be one of the objects that will be dropped by Wario onto the head of either Peach, Mario, or Yoshi. Wanda the fairy would try to guide them to the end of the game's stage.
They would return in Super Mario World 2: Yoshi's Island where they would appear in some of the levels in the game, usually in a fortress or castle. They will hang on the walls or ceilings and can be knocked down by a throw of an egg. Once they have been knocked down, they will drop coins, or sometimes enemies. Some levels require these Buckets to be dropped into water or lava so that the Yoshi can jump into the bucket and ride across the liquid until they reach their destination. They would return in Yoshi's Island DS.
Finally, the Bucket could be found in Shiver City in Paper Mario. Mario needs the Bucket to place on the top of the hat-less Snowman to gain access to Shiver Mountain alongside using the Scarf on the scarfless snowman.