Photometry and Proper Motions of M, L, and T Dwarfs from the Pan-STARRS1 3$\pi$ Survey.

We present a catalog of 9888 M, L and T dwarfs detected in the Pan-STARRS1 3$\pi$ Survey (PS1), covering three-quarters of the sky. Our catalog contains nearly all known objects of spectral types L0-T2 in the PS1 field, with objects as early as M0 and as late as T9, and includes PS1, 2MASS, AllWISE, and Gaia DR1 photometry. We analyze the different types of photometry reported by PS1, and use two types in our catalog to maximize both depth and accuracy. Using parallaxes from the literature, we construct empirical SEDs for field ultracool dwarfs spanning 0.5-12 $\mu$m. We determine typical colors of M0-T9 dwarfs, and we highlight the distinctive colors of subdwarfs and young objects. Our catalog includes 492 L dwarfs detected in $r_{\rm P1}$, the largest sample of L dwarfs detected at such blue wavelengths. We combine astrometry from PS1, 2MASS, and Gaia DR1 to calculate new proper motions for our catalog. We achieve a median precision of 2.9 mas yr$^{-1}$, a factor of $\approx$3-10 improvement over previous large catalogs. Our catalog contains proper motions for 2405 M6-T9 dwarfs and includes the largest set of homogeneous proper motions for L and T dwarfs published to date, 406 objects for which there were no previous measurements, and 1176 objects for which we improve upon previous literature values. We analyze the kinematics of ultracool dwarfs in our catalog and find evidence that bluer but otherwise generic late-M and L field dwarfs (i.e., not subdwarfs) tend to have higher tangential velocities compared to typical field objects. With the public release of the PS1 data, this survey will continue to be an essential tool for characterizing the ultracool dwarf population.