Supernova explosions produce neutrinos and photons. If neutrinos are massless, they should reach the Earth at the same time as the photons reach the Earth from Supernova explosions. This experiment was tried for SN1987A, but the theory of Supernova eplosions wasn't good enough and there weren't enough neutrinos to get a reliable answer. This situation will definitely improve however. Direct experiments to look for neutrino oscillations in terrestrial accelerators and using the products of cosmic rays interacting with the atmosphere of the Earth.
Work needs to be followed up upon.
Sudbury Neutrino Observatory will be sensitive to all types of neutrinos and thus could also answer this question. Planned for later this decade.