A wild goose chase is a pursuit that is likely to prove pointless and unfruitful, as in “We went on a wild goose chase for the antique store she told us about, but we just couldn't find it.” It may surprise you to learn that it actually has nothing to do with wild geese, despite the name.
This slang term references a type of horse race that was popular in some parts of England in the 16th century. In this race, the pack of horses would follow a leader, often adopting a formation that casually resembled a flock of geese. This was extremely challenging, and bettors often commented that it was difficult to predict the outcome of a wild goose chase, let alone profit from it.
When Shakespeare used it, he meant it in a metaphorical sense, referring to one of Romeo's harebrained plans as a “wild goose chase,” meaning that Romeo was embarking on an adventure that was likely to prove futile. He was referring to this as a situation in which one person sets a path that is difficult to follow, exactly like the lead horse in a wild goose chase. As often happened with colorful words and idioms in Shakespeare's work, the slang term was picked up by the general population.