If we choose at random an integral binary form $f(x, z)$ of fixed degree $d$, what is the probability that the superelliptic curve with equation $C \colon: y^m = f(x, z)$ has a $p$-adic point, or better, points everywhere locally? In joint work with Lea Beneish, we show that the proportion of forms $f(x, z)$ for which $C$ is everywhere locally soluble is positive, given by a product of local densities. By studying these local densities, we produce bounds which are suitable enough to pass to the large $d$ limit. In the specific case of curves of the form $y^3 = f(x, z)$ for a binary form of degree 6, we determine the probability of everywhere local solubility to be 96.94\%, with the exact value given by an explicit infinite product of rational function expressions.