The gauged U (1)L µ−Lτ model can provide for additional contributions to the muon anomalous magnetic moment by means of a loop involving the Z gauge boson. However, the parameter space of such models is severely constrained if one combines the latest muon (g − 2) data with various neutrino experiments, such as neutrino trident production, ν − e and ν − q elastic scattering, etc. In a supersymmetric U (1)L µ−Lτ model, a larger region of parameter space opens up, thus enabling one to explore otherwise forbidden regions of parameter space in nonsupersymmetric models involving the new gauge coupling (gX ) and the mass of the Z gauge boson (M Z ) . We show that the minimal model with the minimal supersymmetric Standard Model (MSSM) field content is strongly disfavored from Z-boson decay and neutrino data. We also show that the nonminimal model with two extra singlet superfields can lead to correct neutrino masses and mixing involving both tree-level and one-loop contributions. We find that, in this model, both muon (g − 2) and neutrino data may be simultaneously explained in a parameter region consistent with experimental observations. In addition, we observe that the muon (g − 2) anomaly can be accommodated even with higher values of electroweak sparticle masses compared to the MSSM. Charged lepton-flavorviolating processes (like µ → eγ, τ → µγ, etc.) may have potentially large branching ratios in this scenario. Depending on the magnitude of the supersymmetry contribution to these processes, they may constrain hitherto unconstrained regions of the M Z − gX parameter space. However, we find that these branching fractions never exceed their upper bounds in a region where both muon (g − 2) and neutrino oscillation data can be simultaneously accommodated. arXiv:1805.04415v2 [hep-ph] 5 Nov 2018 c eLµÊ c µLτÊ c τĤuĤd U (1)L µ−Lτ 0 0 0 0 0 1 -1 -1 1 0 0