We introduce a multifractal formalism for potentials defined on shift systems. We prove that the multifractal spectra are a Legendre transform of thermodynamic functions involving the potentials studied. We obtain the fractal distribution of pointwise dimension for g-measures. Such measures are equilibrium states of potentials not necessarily Hölder continuous and generalize Gibbs measures. In connection with phase transition, we also give examples of potentials with a non-unique equilibrium state and non-analytic multifractal spectra.