We study the effect of folding ridges on the scaling properties of randomly crumpled sheets of different kinds of paper in the folded and unfolded states. We found that the mean ridge length scales with the sheet size with the scaling exponent mu determined by the competition between bending and stretching deformations in the folded sheet. This scaling determines the mass fractal dimension of randomly folded balls D{M}=2/mu. We also found that surfaces of crumpled balls, as well as unfolded sheets, both display self-affine invariance with zeta=nu{ph}, if mu < or =nu{ph} , where nu{ph}=34 is the size exponent for crumpled phantom membrane, or both exhibit an intrinsically anomalous roughness characterized by the universal local roughness exponent zeta=0.72+/-0.04 and the material dependent global roughness exponent alpha=mu, when mu>nu{ph}. The physical implications of these findings are discussed.