  Given a non-quasianalytic subadditive weight function $\omega$ we consider
the weighted Schwartz space $\mathcal{S}_\omega$ and the short-time Fourier
transform on $\mathcal{S}_\omega$, $\mathcal{S}'_\omega$ and on the related
modulation spaces with exponential weights. In this setting we define the
$\omega$-wave front set $WF'_\omega(u)$ and the Gabor $\omega$-wave front set
$WF^G_\omega(u)$ of $u\in\mathcal{S}'_{\omega}$, and we prove that they
coincide. Finally we look at applications of this wave front set for operators
of differential and pseudo-differential type.
