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For spaces of constant, linear, and quadratic splines of maximal smoothnesson the Powell-Sabin 12-split of a triangle, the so-called S-bases were recentlyintroduced. These are simplex spline bases with B-spline-like properties on the12-split of a single triangle, which are tied together across triangles in aB\'ezier-like manner.  In this paper we give a formal definition of an S-basis in terms of certainbasic properties. We proceed to investigate the existence of S-bases for theaforementioned spaces and additionally the cubic case, resulting in anexhaustive list. From their nature as simplex splines, we derive simpledifferentiation and recurrence formulas to other S-bases. We establish aMarsden identity that gives rise to various quasi-interpolants and domainpoints forming an intuitive control net, in terms of which conditions for$C^0$-, $C^1$-, and $C^2$-smoothness are derived.

B-spline-like bases for $C^2$ cubics on the Powell-Sabin 12-split