We consider isolated complex surface singularities which can be modelled as quotients of the complex plane by a finite subgroup of SU(2). In Riemannian geometry, these spaces may be viewed as orbifold limits of Ricci-flat Kähler spaces, which are flat away from the singular point. This talk will guide through some of the geometric properties of these spaces close to the singular point, combining tools from Riemannian geometry with the theory of modular forms.