  We study the constraints imposed by neutrino oscillation experiments on the
minimal extension of the Standard Model that can explain neutrino masses, which
requires the addition of just two singlet Weyl fermions. The most general
renormalizable couplings of this model imply generically four massive neutrino
mass eigenstates while one remains massless: it is therefore a minimal 3+2
model. The possibility to account for the confirmed solar, atmospheric and
long-baseline oscillations, together with the LSND/MiniBooNE and reactor
anomalies is addressed. We find that the minimal model can fit oscillation data
including the anomalies better than the standard $3\nu$ model and similarly to
the 3+2 phenomenological models, even though the number of free parameters is
much smaller than in the latter. Accounting for the anomalies in the minimal
model favours a normal hierarchy of the light states and requires a large
reactor angle, in agreement with recent measurements. Our analysis of the model
employs a new parametrization of seesaw models that extends the Casas-Ibarra
one to regimes where higher order corrections in the light-heavy mixings are
significant.
