  We consider an extension of the Standard Model with two singlet leptons, with
masses in the electroweak range, that induce neutrino masses via the see-saw
mechanism, plus a generic new physics sector at a higher scale, $\Lambda$. We
apply the minimal flavor violation (MFV) principle to the corresponding
Effective Field Theory ($\nu$SMEFT) valid at energy scales $E \ll \Lambda$. We
identify the irreducible sources of lepton flavor and lepton number violation
at the renormalizable level, and apply the MFV ans\"atz to derive the scaling
of the Wilson coefficients of the $\nu$SMEFT operators up to dimension six. We
highlight the most important phenomenological consequences of this hypothesis
in the rates for exotic Higgs decays, the decay length of the heavy neutrinos,
and their production modes at present and future colliders. We also comment on
possible astrophysical implications.
