The four- and five-state ferromagnetic Potts models are studied on both the periodic square lattice and the quasiperiodic Penrose lattice via the microcanonical simulation. Results presented here indicate that the four-state Potts model undergoes a second-order phase transition within the same universality class for both the periodic and quasiperiodic cases. Our simulation data are observed to be sensitive to the expected logarithmic corrections, agreement is found with the nonuniversal exact results in the periodic case, and new values are given in the quasiperiodic case. The five-state Potts model is determined to be first order in both cases, and agreement is found with exact results for the periodic case. © 1989 The American Physical Society.