blind alley

Etymology
The notion of ‘blindness’ comes from the lack of a through passage (the ‘eye’). Attested since 1583, and used figuratively since the mid-19th century.

Noun

 * 1)  A street or passageway that leads nowhere.
 * 2)  A course of inquiry that leads nowhere.

Synonyms

 * cul-de-sac
 * dead end