  A systematic geometric theory for the ultradifferentiable (non-quasianalytic
and quasianalytic) wavefront set similar to the well-known theory in the
classic smooth and analytic setting is developed. In particular an analogue of
Bony's Theorem and the invariance of the ultradifferentiable wavefront set
under diffeomorphisms of the same regularity is proven using a Theorem of
Dyn'kin about the almost-analytic extension of ultradifferentiable functions.
Furthermore we prove a microlocal elliptic regularity theorem for operators
defined on ultradifferentiable vector bundles. As an application we show that
Holmgren's theorem and several generalizations hold for operators with
quasianalytic coefficients.
