  In this paper we study the Coleman-Oort conjecture for superelliptic curves,
i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable
polynomial. We prove that up to isomorphism there are at most finitely many
superelliptic curves of fixed genus $g\geq 8$ with CM Jacobians. The proof
relies on the geometric structures of Shimura subvarieties in Siegel modular
varieties and the stability properties of Higgs bundles associated to fibred
surfaces.
