  We propose models for neutrino masses and mixing in the framework of low
scale $U(1)_{L_\mu-L_\tau}$ gauge extension of the standard model. The models
are designed to spontaneously break $U(1)_{L_\mu-L_\tau}$ so that the
$U(1)_{L_\mu-L_\tau}$ gauge boson acquires an MeV scale mass, which is required
to solve the long-standing problem of muon anomalous magnetic moment. Tiny
neutrino masses are obtained by simultaneously invoking the linear and the
inverse seesaw mechanism, and we succeed in realizing two types of one-zero
textures in the active neutrino mass matrix. Both of the obtained textures
favor inverted neutrino mass ordering and are testable in next generation
experiments of neutrinoless double beta decay. We also show that some of extra
scalar bosons can have MeV scale masses and would have significant impacts on
observations of high energy cosmic neutrinos.
