Lepton flavor violation with light vector bosons

New sub-GeV vector bosons with couplings to muons but not electrons have been discussed in order to explain the muon's magnetic moment, the gap of high-energy neutrinos in IceCube or the proton radius puzzle. If such a light Z′Z′ not only violates lepton universality but also lepton flavor, as expected for example from the recent hint for h→μτh→μτ at CMS, the two-body decay mode τ→μZ′τ→μZ′ opens up and for MZ′<2mμMZ′<2mμ gives better constraints than τ→3μτ→3μ already with 20-year-old ARGUS limits. We discuss the general prospects and motivation of light vector bosons with lepton-flavor-violating couplings.