  Let (X, D) be a projective log canonical pair. We show that for any natural
number p, the sheaf (Omega_X^p(log D))^** of reflexive logarithmic p-forms does
not contain a Weil divisorial subsheaf whose Kodaira-Iitaka dimension exceeds
p. This generalizes a classical theorem of Bogomolov and Sommese.
  In fact, we prove a more general version of this result which also deals with
the orbifoldes g\'eom\'etriques introduced by Campana. The main ingredients to
the proof are the extension theorem of Greb-Kebekus-Kov\'acs-Peternell, a new
version of the Negativity lemma, the Minimal Model Program, and a residue map
for symmetric differentials on dlt pairs.
  We also give an example showing that the statement cannot be generalized to
spaces with Du Bois singularities. As an application, we give a
Kodaira-Akizuki-Nakano-type vanishing result for log canonical pairs which
holds for reflexive as well as for K\"ahler differentials.
