  We prove that for almost every initial data $(u_0,u_1) \in H^s \times
H^{s-1}$ with $s > \frac{p-3}{p-1}$ there exists a global weak solution to the
supercritical semilinear wave equation $\partial _t^2u - \Delta u
+|u|^{p-1}u=0$ where $p>5$, in both $\mathbb{R}^3$ and $\mathbb{T}^3$. This
improves in a probabilistic framework the classical result of Strauss who
proved global existence of weak solutions associated to $H^1 \times L^2$
initial data. The proof relies on techniques introduced by T. Oh and O.
Pocovnicu based on the pioneer work of N. Burq and N. Tzvetkov. We also improve
the global well-posedness result of C. Sun and B. Xia for the subcritical
regime $p<5$ to the endpoint $s=\frac{p-3}{p-1}$.
