We obtain general trace formulae in the case of perturbation of self-adjoint operators by self-adjoint operators of class S_m, where m is a positive integer. In [25] a trace formula for operator Taylor polynomials was obtained. This formula includes the Lifshits-Krein trace formula in the case m = 1 and the Koplienko trace formula in the case m = 2. We establish most general trace formulae in the case of perturbation of Schatten-von Neumann class S_m. We also improve the trace formula obtained in [25] for operator Taylor polynomials and prove it for arbitrary functions in the Besov space B_∞1^m(R). We consider several other special cases of our general trace formulae. In particular, we establish a trace formula for m-th order operator differences.