We investigate the components determining bigness of the cotangent bundle \(\Omega ^1_X\) of smooth models X in the birational class of an orbifold surface of general type Y, with a focus on the contribution given by the singularities of Y. A criterion for bigness of \(\Omega _X^1\) is given involving only topological and singularity data on Y. We single out a special case, the Canonical Model Singularities (CMS) criterion, when Y is the canonical model of . We study the singularity invariants appearing in the criterion and determine them for \(A_n\) singularities. Knowledge of these invariants for \(A_n\) singularities allows one to evaluate the \((c_2,c^2_1)\) -geographical range of the CMS criterion and compare it to other criteria. We obtain new examples of resolutions X of hypersurfaces \(Y\subset \mathbb {P}^3\) (with lower degrees) and of cyclic covers Y of \(\mathbb {P}^2\) branched along line arrangements with \(\Omega ^1_X\) big.
