In isogeometric analysis, classical two-dimensional model domains with corners, such as circular sectors or L-shaped domains, can be conveniently represented with a single patch by collapsing one edge of the parametric square into a conical point. However, such parameterizations suffer from a lack of regularity at the polar point, making standard isogeometric approximation theory inapplicable. In this talk, we introduce a novel framework for deriving error estimates for isogeometric analysis on polar domains with corners. To this end, we define polar function spaces on the parametric domain and construct a modified projector on the space of $C^0$-smooth polar splines. To address the corner singularity of the PDE solution, we propose a graded mesh refinement strategy that ensures optimal convergence with appropriate grading parameters. Our approach maintains the tensor-product structure of splines, unlike other local refinement methods in IGA, as the meshes are graded toward the polar point. All theoretical findings are validated through a series of numerical experiments. Finally, we introduce a new isogeometric collocation-based approach that is built on the ideas of our Galerkin framework. A graded collocation scheme is developed, and numerical results are presented to demonstrate its efficiency.