Bigness of the cotangent bundle is a negativity property of the curvature which has important complex analytic consequences, such as on the Kobayashi hyperbolicity properties and the GGL-conjecture for surfaces. We present a birational criterion for a surface to have big cotangent bundle that takes in account the singularities present in the minimal model and describe how it improves upon other criterions. The criterion allows certain geographic regions of surfaces of general type to have big cotangent bundle, that other criterions can not reach. In this spirit, we produce the examples with the lowest slope $c_1^2/c_2$ having big cotangent bundle that are currently known.