  We illustrate the composition properties for an extended family of SG Fourier
integral operators. We prove continuity results for operators in this class
with respect to $L^2$ and weighted modulation spaces, and discuss continuity on
$\mathscr{S}$, $\mathscr{S}^\prime$ and on weighted Sobolev spaces. We study
mapping properties of global wave-front sets under the action of these Fourier
integral operators. We extend classical results to more general situations. For
example, there are no requirements of homogeneity for the phase functions.
Finally, we apply our results to the study of of the propagation of
singularities, in the context of modulation spaces, for the solutions to the
Cauchy problems for the corresponding linear hyperbolic operators.
