In the era of large photometric surveys, the importance of automated and accurate classification is rapidly increasing. Specifically, the separation of resolved and unresolved sources in astronomical imaging is a critical initial step for a wide array of studies, ranging from Galactic science to large scale structure and cosmology. Here, we present our method to construct a large, deep catalog of point sources utilizing Pan-STARRS1 (PS1) 3π survey data, which consists of ∼3 × 109 sources with m 23.5 mag. We develop a supervised machine-learning methodology, using the random forest (RF) algorithm, to construct the PS1 morphology model. We train the model using ∼5 × 104 PS1 sources with HST COSMOS morphological classifications and assess its performance using ∼4 × 106 sources with Sloan Digital Sky Survey (SDSS) spectra and ∼2 × 108 Gaia sources. We construct 11 "white flux" features, which combine PS1 flux and shape measurements across five filters, to increase the signal-to-noise ratio relative to any individual filter. The RF model is compared to three alternative models, including the SDSS and PS1 photometric classification models, and we find that the RF model performs best. By number the PS1 catalog is dominated by faint sources (m 21 mag), and in this regime the RF model significantly outperforms the SDSS and PS1 models. For time-domain surveys, identifying unresolved sources is crucial for inferring the Galactic or extragalactic origin of new transients. We have classified ∼1.5 × 109 sources using the RF model, and these results are used within the Zwicky Transient Facility real-time pipeline to automatically reject stellar sources from the extragalactic alert stream.