  We establish a semiclassical trace formula in a general framework of
microhyperbolic hermitian systems of $h$-pseudodifferential operators, and
apply it to the study of the spectral shift function associated to a pair of
selfadjoint Schr\"odinger operators with matrix-valued potentials. We give Weyl
type semiclassical asymptotics with sharp remainder estimate for the spectral
shift function, and, under the existence of a scalar escape function, a full
asymptotic expansion in the strong sense for its derivative. A time-independent
approach enables us to treat certain potentials with energy-level crossings.
