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.