Let G be a compact Lie group of dimension n. In this work we characterise the membership of classical pseudo-differential operators on G in the trace class ideal & Sscr;(1)(L-2(G)) , as well as in the setting of the Schatten ideals & Sscr;(r) (L-2(G)) , for all r > 0. In particular, we deduce Schatten characterizations of elliptic pseudo-differential operators of (rho,delta)-type for the large range 0 <= delta < rho <= 1. Additional necessary and sufficient conditions are given in terms of the matrix-valued symbols of the operators, which are global functions on the phase space G x G, with the momentum variables belonging to the unitary dual G of G. In terms of the parameters ( rho , delta ), on the torus T (n), we demonstrate the sharpness of our results showing the existence of atypical operators in the exotic class Psi(-& varkappa;)(0,0)(T-n), kappa > 0, belonging to all the Schatten ideals. Additional order criteria are given in the setting of classical pseudo-differential operators. We present also some open problems in this setting.