The existence of neutrino oscillations calls for an extension of the Standard Model that would generate neutrino masses and mixing. We focused for these studies on a typical low-scale seesaw mechanism, the inverse seesaw, and its supersymmetric realization. New fermionic gauge singlets around the TeV scale with potentially large couplings lead to a rich phenomenology. After introducing the models considered, I will present their predictions for for lepton flavour violating (LFV) observables in light of the recent CMS excess in the Higgs to tau mu channel and future experimental sensitivities to LFV lepton decays.