  Neutrinos can be pseudo-Dirac in nature -- Majorana fermions behaving as
Dirac fermions for all practical purposes. In such a scenario, active and
sterile neutrinos are quasi-degenerate in mass, and hence oscillations between
the two, due to their tiny mass-squared difference $(\delta m^2)$, can develop
only over very long baselines. Under this hypothesis, we analyze the neutrino
data from SN1987A, and find a mild preference for a non-zero mass-squared
difference. The same data can also be used to exclude values of $\delta m^2\sim
10^{-20}{\rm eV}^2$ - the smallest constrained so far. We also discuss how
next-generation experiments like the DUNE and Hyper-Kamiokande can probe this
scenario for a future galactic supernova.
