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Given a non-quasianalytic subadditive weight function $\omega$ we considerthe weighted Schwartz space $\mathcal{S}_\omega$ and the short-time Fouriertransform on $\mathcal{S}_\omega$, $\mathcal{S}'_\omega$ and on the relatedmodulation 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 theycoincide. Finally we look at applications of this wave front set for operatorsof differential and pseudo-differential type.

The Gabor wave front set in spaces of ultradifferentiable functions