  We survey a few minimal scalar extensions of the standard electroweak model
that provide a simple setup for massive neutrinos in connection with an
invisible axion. The presence of a chiral $U(1)$ \`a la Peccei-Quinn drives the
pattern of Majorana neutrino masses while providing a dynamical solution to the
strong CP problem and an axion as a dark matter candidate. We paradigmatically
apply such a renormalizable framework to type-II seesaw and to two viable
models for neutrino oscillations where the neutrino masses arise at one and two
loops, respectively. We comment on the naturalness of the effective setups as
well as on their implications for vacuum stability and electroweak
baryogenesis.
