  The gauged $U(1)_{L_{\mu}-L_{\tau}}$ model can provide for additional
contributions to the muon anomalous magnetic moment by means of a loop
involving the $Z^{\prime}$ 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, $\nu
-e$ and $\nu -q$ elastic scattering, etc. In a supersymmetric
$U(1)_{L_{\mu}-L_{\tau}}$ 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 ($g_X$) and the mass
of the $Z^\prime$ gauge boson ($M_{Z^{\prime}}$) . 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-flavor-violating processes (like
$\mu\rightarrow e\gamma$, $\tau\rightarrow \mu\gamma$, 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 $M_{Z^{\prime}}-g_X$ 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.
