Curves with good reduction over a small set of primes often present both unusual arithmetic and geometry. However, they can be difficult to find explicitly in practice. Motivated by earlier work of Nigel Smart, we determine all Picard curves over Q with good reduction away from 3. Picard curves are cyclic trigonal covers of P^1 of genus 3; they are also the simplest example of non-hyperelliptic curves. We explain a correspondence between such isomorphism classes of such curves and certain equivalence classes of binary forms. If time permits, we will discuss connections between the arithmetic of the Jacobians of such curves and a long-standing question of Ihara on the nature of the canonical outer Galois representation associated to P^1 minus three points.