  We study the conductor of Picard curves over $\mathbb{Q}$, which is a product
of local factors. Our results are based on previous results on stable reduction
of superelliptic curves that allow to compute the conductor exponent $f_p$ at
the primes $p$ of bad reduction. A careful analysis of the possibilities of the
stable reduction at $p$ yields restrictions on the conductor exponent $f_p$. We
prove that Picard curves over $\mathbb{Q}$ always have bad reduction at $p=3$,
with $f_3\geq 4$. As an application we discuss the question of finding Picard
curves with small conductor.
