  We generalize the Bogomolov-Gieseker inequality for semistable coherent
sheaves on smooth projective surfaces to smooth Deligne-Mumford surfaces. We
work over positive characteristic $p>0$ and generalize Langer's method to
smooth Deligne-Mumford stacks. As applications we obtain the Bogomolov
inequality for semistable coherent sheaves on a Deligne-Mumford surface in
characteristic zero, and the Bogomolov inequality for semistable sheaves on a
root stack over a smooth surface which is equivalent to the Bogomolov
inequality for the rational parabolic sheaves on a smooth surface $S$. In a
joint appendix with Hao Max Sun, we generalize the Bogomolov inequality formula
to Simpson Higgs sheaves on tame Deligne-Mumford stacks.
