We study weakly order preserving circle maps with a flat interval, which are differentiable even on the boundary of the flat interval. We calculate the Hausdorff dimension of the non-wandering set and we find a sharp transition from degenerate geometry to bounded geometry depending on the degree of the singularities at the boundary of the flat interval. We prove that the non-wandering set has zero Hausdorff dimension in the case of degenerate geometry and it has Hausdorff dimension strictly greater than zero in the case of bounded geometry. This last result is used to prove an important property of Cherry fows; the Hausdorff dimension of the quasi-minimal set is strictly greater than 1.