The atomic mass relations among the mass triplet 96Zr, 96Nb, and 96Mo have been determined by means of high-precision mass measurements using the JYFLTRAP mass spectrometer at the IGISOL facility of the University of Jyväskylä. We report Q values for the 96Zr single and double β decays to 96Nb and 96Mo, as well as the Q value for the 96Nb single β decay to 96Mo, which are Qβð96ZrÞ ¼ 163.96ð13Þ, Qββð96ZrÞ ¼ 3356.097ð86Þ, and Qβð96NbÞ ¼ 3192.05ð16Þ keV. Of special importance is the 96Zr single β-decay Q value, which has never been determined directly. The single β decay, whose main branch is fourfold unique forbidden, is an alternative decay path to the 96Zr ββ decay, and its observation can provide one of the most direct tests of the neutrinoless ββ-decay nuclear-matrix-element calculations, as these can be simultaneously performed for both decay paths with no further assumptions. The theoretical single β-decay rate has been re-evaluated using a shell-model approach, which indicates a 96Zr single β-decay lifetime within reach of an experimental verification. The uniqueness of the decay also makes such an experiment interesting for an investigation into the origin of the quenching of the axial-vector coupling constant gA.