A standard approach to the construction of smooth low degree polynomial splines over an unstructured triangulation is based on splitting of triangles in such a way that the refined triangulation allows the imposition of smoothness constraints without dependence on geometry. A well-established splitting technique is the Powell-Sabin 6-refinement, which can be used to define C1 quadratic splines as well as splines of higher degree and smoothness. In this talk we review the construction of splines over a Powell-Sabin 6-refinement with a special emphasis on C1 cubic splines. We present B-spline-like functions that enjoy favorable properties such as local support, stability, nonnegativity, and a partition of unity. In particular, we discuss what super-smoothness properties these functions possess and how they depend on geometric properties of the underlying refinement. Based on this we explain how to establish approximation spaces that are suitable for completely unstructured triangulations, partially structured triangulations, and triangulations with a high level of symmetry, e.g., three-directional triangulations. This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).