A Wild Goose Chase
A search that has no hope of success (informal)
Example:The police were sent on a wild goose chase looking for the money that the criminals had already spent.
Origin: Many common idioms like this one come from the works of Shakespeare (1564-1616), the famous English poet and dramatist who wrote 38 plays, including Romeo and Juliet and Hamlet.
Mercutio : If thy wits run the wild goose chase, I have don. ( Romeo and Juliet)