  The Standard Model augmented by the presence of gauge-singlet right-handed
neutrinos proves to be an ideal scenario for accommodating nonzero neutrino
masses. Among the new parameters of this ``New Standard Model'' are
right-handed neutrino Majorana masses M. Theoretical prejudice points to M much
larger than the electroweak symmetry breaking scale, but it has recently been
emphasized that all M values are technically natural and should be explored.
Indeed, M around 1-10 eV can accommodate an elegant oscillation solution to the
LSND anomaly, while other M values lead to several observable consequences. We
consider the phenomenology of low energy seesaw scenarios with M less than and
equal to approximately 1 keV. By exploring such a framework with three
right-handed neutrinos, we can consistently fit all oscillation data --
including those from LSND -- while partially addressing several astrophysical
puzzles, including anomalous pulsar kicks, heavy element nucleosynthesis in
supernovae, and the existence of warm dark matter. Furthermore, low-energy
seesaws -- regardless of their relation to the LSND anomaly -- can also be
tested by future tritium beta-decay experiments, neutrinoless double-beta decay
searches, and other observables. We estimate the sensitivity of such probes to
M.
