Abstract We study massive superparticles with Siegel symmetry as toy models of the Green-Schwarz string, and as mechanics actions furnishing minimal representations of supersymmetry upon quantisation. A general construction yielding all such theories is given, and it is shown that the simplest class can be derived from massless models in higher dimensions. We consider the quantisation of massive superparticles using the canonical methods of Dirac, and show that the first-class and second-class constraints can always be defined so as to respect all global symmetries, giving a completely covariant method of quantisation. (This cannot be done in the massless case.) The quantisation is carried out explicitly for the simplest class of models: Dirac brackets are computed and all necessary consistency checks performed. We prove that no operator-ordering problems arise by finding a set of variables for which the covariant Dirac brackets become free canonical graded commutation relations. By considering features shared by massive superparticles and the Green-Schwarz string (but not present for massless superparticles) we argue that the latter should admit a similar covariant quantisation.