  The discovery of non-zero neutrino masses invites one to consider decays of
heavier neutrinos into lighter ones. We investigate the impact of two-body
decays of neutrinos on the neutronization burst of a core-collapse supernova --
the large burst of $\nu_e$ during the first 25 ms post core-bounce. In the
models we consider, the $\nu_e$, produced mainly as a $\nu_3\,(\nu_2)$ in the
normal (inverted) mass ordering, are allowed to decay to $\nu_1\,(\nu_3)$ or
$\bar{\nu}_1\,(\bar{\nu}_3)$, and an almost massless scalar. These decays can
lead to the appearance of a neutronization peak for a normal mass ordering or
the disappearance of the same peak for the inverted one, thereby allowing one
mass ordering to mimic the other. Simulating supernova-neutrino data at the
Deep Underground Neutrino Experiment (DUNE) and the Hyper-Kamiokande (HK)
experiment, we compute their sensitivity to the neutrino lifetime. We find
that, if the mass ordering is known, and depending on the nature of the Physics
responsible for the neutrino decay, DUNE is sensitive to lifetimes $\tau/m
\lesssim 10^6$ s/eV for a galactic SN sufficiently close-by (around 10 kpc),
while HK is sensitive to lifetimes $\tau/m \lesssim 10^7$ s/eV. These
sensitivities are far superior to existing limits from solar-system-bound
oscillation experiments. Finally, we demonstrate that using a combination of
data from DUNE and HK, one can, in general, distinguish between decaying Dirac
neutrinos and decaying Majorana neutrinos.
