We discuss the science motivations and prospects for a joint analysis of gravitational wave (GW) and low-energy neutrino data to search for prompt signals from nearby supernovae (SNe). Both gravitational wave and low-energy neutrinos are expected to be produced in the innermost region of a core-collapse supernova, and a search for coincident signals would probe the processes which power a supernova explosion. It is estimated that the current generation of neutrino and gravitational wave detectors would be sensitive to galactic core-collapse supernovae, and would also be able to detect electromagnetically dark SNe. A joint GW-neutrino search would enable improvements to searches by way of lower detection thresholds, larger distance range, better live-time coverage by a network of GW and neutrino detectors, and increased significance of candidate detections. A close collaboration between the GW and neutrino communities for such a search will thus go far toward realizing a much sought-after astrophysics goal of detecting the next nearby supernova.