  While three flavors of "active" neutrinos are consistent with mixing angle
results within error bars, there are anomalies may be hints of physics beyond
the standard model that can accommodate a fourth mostly "sterile" neutrino
species with an eV-scale mass and a mixing angle with active neutrinos of order
$\theta_0\simeq 0.1$. We describe a scenario with eV-scale sterile neutrinos
that have self-interactions via a new gauge vector boson, a "secret" mediator
$\phi$. We show that their production in the early Universe via mixing with
active neutrinos is consistent with Big Bang Nucleosynthesis and free-streaming
constraints in the Cosmic Microwave Background epoch. For $M_\phi=4-8$ keV and
sterile neutrino coupling $g_s=10^{-4}$, we find that resonant interactions of
diffuse supernova neutrinos with relic sterile neutrinos in transit to the
Earth would cause spectral dips in the neutrino flux. We illustrate the
corresponding (anti-)neutrino event distributions as a function of energy in
the DUNE (Hyper-Kamiokande) detector.
