What it means:
complicated and fruitless pursuit or search
How Shakespeare used it:
In Romeo and Juliet, Mercutio likens the rapid exchange of jokes between Romeo and himself to the cross-country horse race of Shakespeare's time, known as the wild goose chase, in which any number of riders tried to keep up with and accurately follow the lead rider's course:
"Nay, if our wits run the wild-goose chase, I am done; for thou / hast more of the wild goose in one of thy wits than, I am sure, I / have in my whole five." (Act 2, Scene 4)
The name of the race derives from its similarity to the flight of geese in formation, with a group of geese following behind a leader that sets the course.
"Seriously just went on a wild goose chase for a place to study. Everything is packed so I found an empty classroom in the math building." TheHeartquake on Twitter, May 11, 2009