If we choose at random a polynomial $f(x)$ of fixed degree $d$, how likely is the superelliptic curve $C \colon: y^m = f(x)$ to have a $p$-adic point? What about points everywhere locally? In joint work with Lea Beneish, we show that the proportion of polynomials $f(x)$ for which $C$ is everywhere locally soluble is positive, given by a product of local densities. By studying these local densities, we produce bounds suitable to pass to the large $d$ limit. In the case of the family of genus 4 curves of the form $y^3 = f(x,z)$ for a binary form of degree 6, we compute this probability exactly, given as an explicit infinite product and approximately equal to 97%.