  If the heaviest neutrino mass eigenstate is unstable, its decay modes could
include lighter neutrino eigenstates. In this case part of the decay products
could be visible, as they would interact at neutrino detectors via mixing. At
neutrino oscillation experiments, a characteristic signature of such
\emph{visible neutrino decay} would be an apparent excess of events at low
energies. We focus on a simple phenomenological model in which the heaviest
neutrino decays as $\nu_3 \rightarrow \nu_{1,2} + \phi$, where $\phi$ is a new
light scalar. If neutrinos are Majorana particles the helicity-flipping decays
would be observable (i.e., $\nu \to \bar\nu + \phi$), leading to interesting
observable consequences on the event rates. We compute the sensitivities of the
Deep Underground Neutrino Experiment (DUNE) to the couplings of the new scalar
as a function of the lightest neutrino mass. Under the assumption that only the
heaviest neutrino is unstable, and for a normal mass ordering, we find that
DUNE will be sensitive to values of $\tau_3/m_3 > 1.95 - 2.6\times
10^{-10}$~s/eV (90\% C.L.) (depending on the lightest neutrino mass), where
$\tau_3$ and $m_3$ are the lifetime and mass of $\nu_3$, respectively.
