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.