We study the local geometry of smooth curves on singular surfaces. This amounts to the following problem in local algebra: Classify pairs \((R,p)\) consisting of a complete local \(\mathbb{C}\)-algebra \(R\), together with a height one prime ideal \(p\) of \(R\) such that \(R/p\) is regular. It is assumed that \(R\) is the complete local ring of a singular point on a variety. Two pairs \((R,p)\) and \((R',p')\) are isomorphic if there exists an isomorphism of \(\mathbb{C}\)-algebras from \(R\) to \(R'\) which carries \(p\) to \(p'\). A complete solution is given in the special case where \(R\) corresponds to a rational double point.