  Ever since the discovery of neutrinos, we have wondered if neutrinos are
their own antiparticles. One remarkable possibility is that neutrinos have a
pseudo-Dirac nature, predicting a tiny mass difference between active and
sterile states. We analyze the neutrino data from SN1987A in the light of
active-sterile oscillations and find a mild preference ($\Delta\chi^2\approx
3$) for $\delta m^2=6.31\times 10^{-20}{\rm eV}^2$. Notably, the same data is
able to exclude $\delta m^2\sim[2.55,3.01]\times 10^{-20}{\rm eV}^2$ with
$\Delta\chi^2> 9$, the tiniest mass differences constrained so far. We further
consider the next-generation of experiments and demonstrate their sensitivity
exploring the nature of the neutrino mass.
