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' not only violates lepton universality but also lepton flavor, as expected for example from the recent hint for $h\to\mu\tau$ at CMS, the two-body decay mode $\tau \to \mu Z'$ opens up and for $M_{Z'} < 2 m_\mu$ gives better constraints than $\tau\to 3\mu$ already with 20-year-old ARGUS limits. We discuss the general prospects and motivation of light vector bosons with lepton-flavor-violating couplings.