The resonance patterns and lasing modes in a spiral-shaped dielectric microcavity are investigated through passive and active medium calculations. We find that the high-Q resonance modes are whispering-gallery-like modes, and these resonance modes can be easily excited as lasing modes. We also find that the quasi-scarred resonance mode, which shows strong directional emission beams from the cavity boundary, can be excited with selectively applied external pumping. Through a spectral analysis of the time evolution of the light field, the competition between these lasing modes is discussed.