In this paper, a slot SiC waveguide with four zero-dispersion wavelengths (ZDWs) is proposed for the supercontinuum (SC) and frequency comb generations. The four ZDWs are located at 1089, 1656, 2222, and 3113nm, respectively. The effects of the pump wavelength, peak power, pulse width, and waveguide length on the SC generation are discussed when the waveguide is pumped near the first ZDW and the third ZDW in the anomalous dispersion region, respectively. While pumping at 1300 and 2450nm, a near-infrared SC covering from 873 to 2444nm and a mid-infrared SC extending from 1349 to 4727nm with good coherences are obtained, respectively. By utilizing a 50-pulses pump source at a repetition rate of 1GHz, we also investigate the SC-based frequency comb generation.