  The main goal of this paper is to prove the polystability of the logarithmic
tangent sheaf $\mathscr T_X(-D)$ of a log canonical pair $(X,D)$ whose
canonical bundle $K_X+D$ is ample, generalizing in a significant way a theorem
of Enoki. We apply this result and the techniques involved in its proof to get
a version of this theorem for stable varieties (the higher dimensional analogue
of Deligne-Mumford's stable curves) and to prove the polystability with respect
to any polarization of the tangent sheaf of a singular Calabi-Yau variety.
