  We discuss all possible schemes with four massive neutrinos inspired by the
existing experimental indications in favour of neutrino mixing, namely the
atmospheric, solar and LSND neutrino experiments. We argue that the scheme with
a neutrino mass hierarchy is not compatible with the experimental results,
likewise all other schemes with the masses of three neutrinos close together
and the fourth mass separated by a gap needed to incorporate the LSND neutrino
oscillation result. Only two schemes with two pairs of neutrinos with nearly
degenerate masses separated by this gap of the order of 1 eV are in agreement
with the results of all experiments, including those where no indications for
neutrino oscillations have been found. We also point out the possible effect of
big-bang nucleosynthesis on the 4-neutrino mixing matrix and its consequences
for neutrino oscillations. Finally, we study predictions for neutrino
oscillation experiments and $^3$H and $(\beta \beta)_{0\nu}$ decays, following
from the two favoured neutrino mass spectra and mixing schemes. These
predictions can be conceived as checks of the input used for arriving at the
two favoured schemes.
