  Quasi-analytic wave-front sets of distributions which correspond to the
Gevrey sequence $p!^s$, $s\in[1/2,1)$ are defined and investigated. The
propagation of singularities are deduced by considering sequences of Gaussian
windowed short-time Fourier transforms of distributions which are modifications
of the original distributions by suitable restriction-extension techniques.
Basic micro-local properties of the new wave-fronts are thereafter established.
