The regular model of a curve is a key object in the study of the arithmetic of the curve, as information about the special fiber of a regular model provides information about its generic fiber (such as rational points through the Chabauty-Coleman method, index, Tamagawa number of the Jacobian, etc). Every curve has a somewhat canonical regular model obtained from the quotient of a regular semistable model by resolving only singularities of a special type called quotient singularities. We will discuss in this talk what is known about the resolution graphs of $Z/pZ$-quotient singularities in the wild case, when $p$ is also the residue characteristic. The possible singularities that can arise in this process are not yet completely understood, even in the case of elliptic curves in residue characteristic 2.