Neutrino mass is very hard to observe directly.
In principle, to show that a particle has mass, you just need to show that it travels at less than the speed of light. However, most neutrinos are produced traveling at almost the speed of light, and they interact extremely weakly with all forms of matter, so they are very hard to slow down. Instead, most searches for neutrino mass rely on a quantum-mechanical interference effect called neutrino oscillation. If neutrinos have mass, and if the neutrinos produced by the weak interactions do not have definite mass, but are quantum-mechanical mixtures of states with different mass, then the composition of a beam of neutrinos will oscillate, or change flavor back and forth with distance from the source. For example, it could oscillate from electron neutrino to muon neutrino.