Core-collapse SNe (CCSNe): Systematic searches of radio emission from CCSNe are still lacking, and only targeted searches of radio emission from just some of the optically discovered CCSNe in the local universe have been carried out. Optical searches miss a significant fraction of CCSNe due to dust obscuration; CCSN radio searches are thus more promising for yielding the complete, unobscured star-formation rates in the local universe. The SKA yields the possibility to piggyback for free in this area of research by carrying out commensal, wide-field, blind transient survey observations. SKA1-SUR should be able to discover several hundreds of CCSNe in just one year, compared to about a dozen CCSNe that the VLASS would be able to detect in one year, at most. SKA, with an expected sensitivity ten times that of SKA1, is expected to detect CCSNe in the local Universe by the thousands. Therefore, commensal SKA observations could easily result in an essentially complete census of all CCSNe in the local universe, thus yielding an accurate determination of the volumetric CCSN rate. Type Ia SNe: We advocate for the use of the SKA to search for the putative prompt (first few days after the explosion) radio emission of any nearby type Ia SN, via target-of-opportunity observations. The huge improvement in sensitivity of the SKA with respect to its predecessors will allow to unambiguously discern which progenitor scenario (single-degenerate vs. double-degenerate) applies to them.