In this paper we present symbolic criteria for invariant operators on compact
topological groups $G$ characterising the Schatten-von Neumann classes
$S_{r}(L^{2}(G))$ for all $0<r\leq\infty$. Since it is known that for
pseudo-differential operators criteria in terms of kernels may be less
effective (Carleman's example), our criteria are given in terms of the
operators' symbols defined on the noncommutative analogue of the phase space
$G\times\hat{G}$, where $G$ is a compact topological (or Lie) group and
$\hat{G}$ is its unitary dual. We also show results concerning general
non-invariant operators as well as Schatten properties on Sobolev spaces. A
trace formula is derived for operators in the Schatten class $S_{1}(L^{2}(G))$.
Examples are given for Bessel potentials associated to sub-Laplacians (sums of
squares) on compact Lie groups, as well as for powers of the sub-Laplacian and
for other non-elliptic operators on SU(2)$\simeq\mathbb S^3$ and on SO(3).