Massive neutrinos and leptonic mixings have provided the first evidence of flavour violation in the lepton sector, opening a unique gateway to many new phenomena. Among the latter, one finds processes violating lepton number, charged lepton flavours, or even the universality of lepton flavours. These very rare transitions can be studied in high-intensity facilities, and if observed, constitute a clear sign of New Physics. After a brief review of the experimental status of dedicated searches, we comment on the prospects of several well-motivated models of neutrino mass generation to several of the above mentioned observables, also discussing how the interplay of high-intensity observables and neutrino data can shed light on the underlying New Physics model.