neutrino oscillation
Send an electron neutrino on a long journey and something bizarre happens: partway there, it may arrive as a muon or tau neutrino instead. A neutrino changes its flavor in flight, oscillating between the three types as it travels. This is the phenomenon that solved a thirty-year mystery, why detectors kept counting only about a third as many neutrinos from the Sun as theory demanded, and it is the clearest crack ever found in the Standard Model.
Neutrino oscillation happens because the neutrino's flavor states (electron, muon, tau) are not the same as its mass states (labeled 1, 2, 3); each flavor is a quantum superposition of the three mass states, related by a mixing matrix called the PMNS matrix. The mass states travel at very slightly different rates because they have slightly different masses, so their relative quantum phases drift, and the flavor you would measure changes with distance. For two flavors the survival probability has the schematic form P = 1 - sin^2(2 theta) sin^2(1.27 Delta(m^2) L / E), where theta is a mixing angle, Delta(m^2) the difference of squared masses in eV^2, L the distance in km and E the energy in GeV. The oscillation therefore depends on L over E, and its very existence requires that at least two neutrinos have different, nonzero masses.
That last point is why oscillation matters so much: massless particles cannot oscillate, so the observation of oscillation proves neutrinos have mass, contradicting the original massless-neutrino Standard Model. The solar-neutrino deficit measured at Homestake and resolved by SNO, and the atmospheric-neutrino anomaly measured by Super-Kamiokande, were both flavor oscillation, and the discovery earned the 2015 Nobel Prize. Oscillation experiments now measure the mixing angles and mass-squared differences precisely, but they give only differences of squared masses, not the absolute mass scale, and whether neutrinos violate CP is a frontier question.
Super-Kamiokande found that muon neutrinos produced by cosmic rays in the atmosphere arrived in the expected numbers from directly overhead but were depleted when they came up through the whole Earth; the extra distance had given them time to oscillate into tau neutrinos the detector could not see.
The same neutrinos, depleted only when they travel farther, are oscillation caught in the act.
Oscillation is sensitive only to differences of squared masses (Delta m^2), so it proves neutrinos are massive and unequal in mass but cannot by itself tell you the absolute mass of any neutrino.