Representing arbitrary surfaces with a finite number of polynomial patches requires the introduction of polar points for high-valence neighborhoods in quadrilateral meshes. Such holes can be filled by means of polar spline surfaces, where the basic idea is to use periodic spline patches with one collapsed boundary. Building splines over such singularities requires special rules to ensure smoothness; ensuring suitability for design and analysis imposes further constraints. In this talk, we focus on C^k polar spline parametric patches of arbitrary degree and with arbitrary number of elements at the hole boundary. We present a simple, geometric construction of smooth B-spline basis functions over such polar parametric domains possessing interesting properties as non-negativity and partition of unity. In addition, the constructed spline spaces show optimal approximation behavior, even at the polar singular point. We end with some applications of the technology to free-form modeling and the solution of high-order PDEs, such as the Cahn-Hilliard equations, where the smoothness afforded by the spline basis allows straightforward numerical discretization and implementation.