Hyperelastic constitutive models are investigated and compared on their ability to predict the elastic, isothermal and rate-independent response of rubber. Constitutive model parameters are identified in an optimization problem by minimizing the difference between homogeneous experimental data and their analytical solutions. The results are presented for ten hyperelastic constitutive models over four case studies where varying extents of experimental data are used. The choice of constitutive model is found to determine how accurately experimental data is fitted, though this has different implications depending on the extent of available experimental data. With a complete data set, an accurate fit generally indicates an overall accurate prediction of the material’s response. However, an accurate fit to a reduced set of experimental data may not indicate an accurate prediction of the overall response. With reduced data, accurate predictions are obtained only if the constitutive model is capable of predicting unfitted deformations and the appropriate experimental data is used.