  We study the constraints imposed by neutrino oscillation experiments on the
minimal extensions of the Standard Model (SM) with $n_R$ gauge singlet fermions
("right-handed neutrinos"), that can account for neutrino masses. We consider
the most general coupling to SM fields of the new fields, in particular those
that break lepton number and we do not assume any a priori hierarchy in the
mass parameters. We proceed to analyze these models starting from the lowest
level of complexity, defined by the number of extra fermionic degrees of
freedom. The simplest choice that has enough free parameters in principle (i.e.
two mass differences and two angles) to explain the confirmed solar and
atmospheric oscillations corresponds to $n_R=1$. This minimal choice is shown
to be excluded by data. The next-to-minimal choice corresponds to $n_R=2$. We
perform a systematic study of the full parameter space in the limit of
degenerate Majorana masses by requiring that at least two neutrino mass
differences correspond to those established by solar and atmospheric
oscillations. We identify several types of spectra that can fit long-baseline
reactor and accelerator neutrino oscillation data, but fail in explaining solar
and/or atmospheric data. The only two solutions that survive are the expected
seesaw and quasi-Dirac regions, for which we set lower and upper bounds
respectively on the Majorana mass scale. Solar data from neutral current
measurements provide essential information to constrain the quasi-Dirac region.
The possibility to accommodate the LSND/MiniBoone and reactor anomalies, and
the implications for neutrinoless double-beta decay and tritium beta decay are
briefly discussed.
