It is well known that the spectrum condition, i.e. the positivity of the energy in every inertial coordinate system, is one of the central conceptual ingredients in model-independent approaches to relativistic quantum field theory. When one attempts to formulate quantum field theory in a model-independent manner on a curved background spacetime, it is not immediately clear which concepts replace the spectrum condition. The present work is devoted to reviewing facets of this situation, thereby focussing on one particular approach that attempts to generalize the notion of energy-momentum spectrum by the notion of "wavefront set", which may be seen as an asymptotic high-frequency part of the spectrum.