useless search is correct because wild goose chase means a useless or hopeless search for something difficult or impossible to find.

This idiom or phrase is useful because it helps students understand the intended meaning beyond the individual words.

Examples: Looking for the lost key in the wrong building was a wild goose chase. The false clue sent the detectives on a wild goose chase.

The sentence describes an unsuccessful search, not a useful, timely, or wise action.