We investigated the spectral and coherence signatures of threshold in random lasers with incoherent feedback consisting of alumina colloidal nanoparticles suspended in rhodamine 6G methanol solution under nanosecond-pulsewidth pumping, based on measurement of temporal and spatial coherence properties and comparison with emission spectra. Feedback in this random laser was provided by multiple scattering from the alumina particles, and the effects of particle concentration and scattering length were studied for the weakly scattering and diffusive scattering regimes. At threshold, in each regime, the visibility of the interference fringes jumped abruptly, coinciding with a substantial increase in peak emission intensity and decrease in the linewidth of a single dominant emission peak.