fool's errand

a fool's errand
(an attempt to do something that has no chance of success; foolish undertaking) — бесплодная, глупая, бесполезная, бессмысленная пустая, напрасная затея / трата сил; безнадёжное дело; напрасный труд

Example 1: Billions of dollars have been spent on long-range weather forecasting, but it's a fool's errand. (thefreedictionary.com)
Example 2: Trying to reduce head injuries simply by reducing the number of cyclists is a fool’s errand.

go on a fool's errand

send sb on a fool's errand (to make someone go somewhere or do something for no good reason) — дать кому-л. бессмысленное поручение, послать кого-л. неизвестно за чем (≈ пойди туда — не знаю куда, принеси то — не знаю что)

see also
[wild goose chase]
wasted effort