We define a universal Fréchet set S of a Banach space Y as a subset containing a point of Fréchet differentiability of every Lipschitz function g : Y -> R. We prove a sufficient condition for S to be a universal Fréchet set and use this to construct new examples of such sets. The strongest such result says that in a non-zero Banach space Y with separable dual one can find a universal Fréchet set S ⊆ Y that is closed, bounded and has Hausdorff dimension one.