Corner singularities play a significant role for the modeling of complex physical phenomena in non-smooth domains. Their presence renders simulations challenging as standard methods produce suboptimal results due to singular solutions. In isogeometric analysis, two-dimensional model domains with corner singularities like circular sectors or L-shapes can be discretized conveniently with a single patch. However, the corresponding isogeometric parameterization is singular and standard approximation spaces are not appropriate. Hence, two major challenges need to be tackled at once: the singularity of both the solution and the parameterization which lie at the same corner. We introduce a graded mesh refinement algorithm, enabling locally refined meshes near the singular corner and combine it with modified approximation spaces that have been proposed for singularly parametrized domains in the literature. We prove optimal convergence of our method for solving boundary value and eigenvalue problems on circular sectors. To confirm the theory, we provide numerical results, where we also extend our approach to other domains with corner singularities. Moreover, we consider the robustness of standard isogeometric function spaces with respect to singular parameterizations and investigate the role of the spline regularity.