Abstract Core-collapse supernovae (CCSNe) are predicted to produce gravitational waves (GWs) that may be detectable by Advanced LIGO/Virgo. These GW signals carry information from the heart of these cataclysmic events, where matter reaches nuclear densities. Recent studies have shown that it may be possible to infer the properties of the proto-neutron star (PNS) via GWs generated by hydrodynamic perturbations of the PNS. However, we lack a comprehensive understanding of how these relationships may change with the properties of CCSNe. In this work, we build a self-consistent suite of over 1000 exploding CCSNe from a grid of progenitor masses and metallicities combined with six different nuclear equations of state (EOS). Performing a linear perturbation analysis on each model, we compute the resonant GW frequencies of the PNS, and we motivate a time-agnostic method for identifying characteristic frequencies of the dominant GW emission. From this, we identify two characteristic frequencies, of the early- and late-time signal, that measure the surface gravity of the cold remnant neutron star, and simultaneously constrain the hot nuclear EOS. However, we find that the details of the CCSN model, such as the treatment of gravity or the neutrino transport, and whether it explodes, noticeably change the magnitude and evolution of the PNS eigenfrequencies.