A no-win scenario or no-win situation was a situation in which every possible outcome was undesirable.
The Kobayashi Maru scenario was one example of such a quandary, though James T. Kirk, as well as his alternate reality counterpart, said he didn't believe there was such a thing. (Star Trek II: The Wrath of Khan; Star Trek)
While taking the Bridge officer's test, Deanna Troi was initially frustrated by the engineering qualification, which involved ordering a crew member to their certain death. She eventually passed. (TNG: "Thine Own Self")