Summary: We propose a feasible Kochen-Specker experiment with single photons, whose goal is to submit noncontextual hidden-variable theories to nonstatistical, all-or-nothing tests. Our proposal improves some recently performed tests in two respects. First, the present proposal is state-independent, thereby realizing a scheme designed by Cabello and Garcí-Alcaine that was only partially realized in recent experiments. Second, our proposal can deal with non-ideal detectors without invoking the fair sampling assumption.