  For spaces of constant, linear, and quadratic splines of maximal smoothness
on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently
introduced. These are simplex spline bases with B-spline-like properties on the
12-split of a single triangle, which are tied together across triangles in a
B\'ezier-like manner.
  In this paper we give a formal definition of an S-basis in terms of certain
basic properties. We proceed to investigate the existence of S-bases for the
aforementioned spaces and additionally the cubic case, resulting in an
exhaustive list. From their nature as simplex splines, we derive simple
differentiation and recurrence formulas to other S-bases. We establish a
Marsden identity that gives rise to various quasi-interpolants and domain
points forming an intuitive control net, in terms of which conditions for
$C^0$-, $C^1$-, and $C^2$-smoothness are derived.
