It is well known that for surfaces the positivity property of the cotangent bundle $\Omega^1_X$ called bigness implies hyperbolic properties. We give a criterion for bigness of $\Omega^1_X$ involving the singularities of the canonical model of $X$ and compare it with other criterions. The criterion involves invariants of the canonical singularities whose values were unknown. We describe a method to find the invariants and obtain formulas for the $A_n$ singularities. An application of this work is to determine for which degrees do hypersurfaces in $\mathbb {P}^3$ have deformations with big cotangent bundles and have symmetric differentials of low degrees.