  The space of $C^1$ cubic Clough-Tocher splines is a classical finite element
approximation space over triangulations for solving partial differential
equations. However, for such a space there is no B-spline basis available,
which is a preferred choice in computer aided geometric design and isogeometric
analysis. A B-spline basis is a locally supported basis that forms a convex
partition of unity. In this paper, we explore several alternative $C^1$ cubic
spline spaces over triangulations equipped with a B-spline basis. They are
defined over a Powell-Sabin refined triangulation and present different types
of $C^2$ super-smoothness. The super-smooth B-splines are obtained through an
extraction process, i.e., they are expressed in terms of less smooth basis
functions. These alternative spline spaces maintain the same optimal
approximation power as Clough-Tocher splines. This is illustrated with a
selection of numerical examples in the context of least squares approximation
and finite element approximation for second and fourth order boundary value
problems.
