Isolated singularities of the moduli space for compact Riemann surfaces of genus > 3 (in Zariski's sense) correspond to quasiplatonic curves. These can be caracterised in many different ways, e.g. by the existence of regular dessins or by normal subgroups of Fuchsian triangle groups. The talk will recall some of these caracterisations and report about joint work with J.-Chr. Schlage-Puchta about the growth of the number of these isolated singularities with large genus.