  We construct Bridgeland stability conditions on the derived category of
smooth quasi-projective Deligne-Mumford surfaces whose coarse moduli spaces
have ADE singularities. This unifies the construction for smooth surfaces and
Bridgeland's work on Kleinian singularities. The construction hinges on an
orbifold version of the Bogomolov-Gieseker inequality for slope semistable
sheaves on the stack, and makes use of the To\"en-Hirzebruch-Riemann-Roch
theorem.
