The morphology of laser-induced damage sites at the exit surface of fused silica is tightly correlated to the mode composition of the nanosecond laser pulses at 1064 nm. In the single longitudinal mode (SLM) configuration, a molten and fractured central zone is surrounded by a funnel-shaped surface modification. Ring patterns surround the damage sites when these are initiated by multiple longitudinal modes (MLM) laser pulses. In this last mode configuration, the pulses temporal profiles as well as the damage ring patterns differ from pulse to pulse. The appearance chronology of the rings is found to be closely related to the temporal shape of the laser pulses. This supports that the damage morphology originates from the coupling of a laser-supported detonation wave propagating in air with an ablation mechanism in silica. In our experiments, the propagation speed of the detonation wave reaches about 20 km/s and scales as the cube root of the laser intensity, in good agreement with theory.