We consider the Euler equations on the torus in dimensions $2$ and $3$ and we construct invariant measures for the dynamics of these equations concentrated on sufficiently regular Sobolev spaces so that strong solutions are also known to exist at least locally. The proof follows the method of Kuksin, and we obtain in particular that these measures do not have atoms, excluding trivial invariant measures such as diracs... Then we prove that almost every initial data gives rise to a global solution for which the growth of the Sobolev norms are at most polynomial. (read more)