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) 3$pi$ survey data, which consists of $sim$3$times10^9$ sources with $mlesssim23.5,$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 $sim$5$times10^4$ PS1 sources with HST COSMOS morphological classifications and assess its performance using $sim$4$times10^6$ sources with Sloan Digital Sky Survey (SDSS) spectra and $sim$2$times10^8$ textit{Gaia} sources. We construct 11 white flux features, which combine PS1 flux and shape measurements across 5 filters, to increase the signal-to-noise ratio relative to any individual filter. The RF model is compared to 3 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 ($mgtrsim21,$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 $sim$1.5$times10^9$ 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.
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