The model of local non-Gaussianity, parameterized by the constant non-linearity parameter fNL, is an extremely popular description of non-Gaussianity. However, a mild scale-dependence of fNL is natural. This scale dependence is a new observable, potentially detectable with the Planck satellite, which helps to further discriminate between models of inflation. It is sensitive to properties of the early universe which are not probed by the standard observables. In a complementary way, the trispectrum also contains important information about non-Gaussianity which the bispectrum does not capture. We explicitly calculate the scale dependence and trispectrum in several models including one with a very large infrared-loop contribution to the bispectrum and in various realizations of the curvaton scenario.