Self-pulsing and dual-mode lasing in a square microcavity semiconductor laser are studied experimentally. Self-sustained pulses originating from undamped relaxation oscillation induced by a two-mode interaction are obtained, as the injection current is slightly above the laser threshold. A repetition frequency of 4.4 GHz and a pulse width of 30-40 ps are obtained at a current of 8 mA. The laser switches to continuous-wave operation when the injection current is higher than a certain value, and dual-mode lasing with 30.7 GHz at 16 mA and 10.7 GHz at 27 mA are observed in the lasing spectra. Furthermore, the relative intensity noise spectra are presented to reveal the relationship between the lasing states and the dynamics induced by relaxation oscillation and mode beating.