Uncertainties in our knowledge of the properties of dense ma er near and above nuclear saturation density are among the main sources of variations in multi-messenger signatures predicted for core-collapse supernovae (CCSNe) and the properties of neutron stars (NSs). We construct 97 new nite-temperature equations of state (EOSs) of dense ma er that obey current experimental, observational, and theoretical constraints and discuss how systematic variations in the EOS parameters a ect the properties of cold nonrotating NSs and the core collapse of a 20-M progenitor star. e core collapse of the 20-M progenitor star is simulated in spherical symmetry using the general-relativistic radiation-hydrodynamics code GR1D where neutrino interactions are computed for each EOS using the NuLib library. We conclude that the e ective mass of nucleons at densities above nuclear saturation density is the largest source of uncertainty in the CCSN neutrino signal and dynamics even though it plays a subdominant role in most properties of cold NS ma er. Meanwhile, changes in other observables a ect the properties of cold NSs, while having li le e ect in CCSNe. To strengthen our conclusions, we perform six octant three-dimensional CCSN simulations varying the e ective mass of nucleons at nuclear saturation density. We conclude that neutrino heating and, thus, the likelihood of explosion is signi cantly increased for EOSs where the e ective mass of nucleons at nuclear saturation density is large.PACS numbers: 21.65. Mn,26.50.+x,26.60.Kp