The uniform and non-uniform scaling properties of the exact correlation energy functional are applied in the study of rational, polynomial, logarithmic and exponential local and gradient-dependent forms of the correlation energy functional, which depend separably on the density and on the reduced variable of the gradient of the density. No local or gradient-dependent polynomial, rational, logarithmic, and exponential form can satisfy all the properties at the same time for the high- or low-density limits. A slight violation of some of the conditions may lead to simple approximate expressions for the correlation energy functional.