The simplest wild quotient singularities arise when the group G=Z/pZ acts faithfully on the ring of power series A=k[[u,v]] in characteristic p>0. Such singularities play an important role in various areas, for example reduction of curves or automorphisms of algebraic surfaces. Generalizing work of Artin for p=2, we introduce so-called moderately ramified actions and establish several structure results on the ensuing ring of invariants. Building on this, we prove some results on the possible intersection matrices arising from resolutions of singularities for wild quotients. This is joint work with Dino Lorenzini.