We introduce global wave-front sets ${WF}_{M(\omega,\mathscr B)}(f)$, $f\in
{\mathscr S}^\prime(\mathbf{R}^d)$, with respect to the modulation spaces
$M(\omega,\mathscr B)$, where $\omega$ is an appropriate weight function and
$\mathscr B$ is a translation invariant Banach function space. We show that the
standard properties for known notions of wave-front set extend to
${WF}_{M(\omega,\mathscr B)}(f)$. In particular, we prove that microlocality
and microellipticity hold for a class of globally defined pseudo-differential
operators.

We investigate the lifting property of modulation spaces and construct
explicit isomorpisms between them. For each weight function $\omega$ and
suitable window function $\fy $, the Toeplitz operator (or localization
operator) $\tp_\fy (\omega)$ is an isomorphism from $M^{p,q}_{(\omega_0)}$ onto
$M^{p,q}_{(\omega_0/\omega)}$ for every $p,q \in [1,\infty ]$ and arbitrary
weight function $\omega_0$. The methods involve the pseudo-differential
calculus of Bony and Chemin and the Wiener algebra property of certain symbol
classes of pseudo-differential operators.