The production of a μ+μ- pair from the scattering of a muon neutrino off the Coulomb field of a nucleus, known as neutrino trident production, is a subweak process that has been observed in only a couple of experiments. As such, we show that it constitutes an exquisitely sensitive probe in the search for new neutral currents among leptons, putting the strongest constraints on well-motivated and well-hidden extensions of the standard model gauge group, including the one coupled to the difference of the lepton number between the muon and tau flavor, Lμ-Lτ. The new gauge boson Z′, increases the rate of neutrino trident production by inducing additional (μ̄γ αμ)(ν̄γαν) interactions, which interfere constructively with the standard model contribution. Existing experimental results put significant restrictions on the parameter space of any model coupled to muon number Lμ, and disfavor a putative resolution to the muon g-2 discrepancy via the loop of Z′ for any mass mZ′ 400MeV. The reach to the models' parameter space can be widened with future searches of the trident production at high-intensity neutrino facilities such as the LBNE.