Classifying the morphologies of radio galaxies is important to understand their physical properties and evolutionary histories. A galaxy's morphology is often determined by visual inspection, but as survey size increases robust automated techniques will be needed. Deep neural networks are an attractive method for automated classification, but have many free parameters and therefore require extensive training data and are subject to overfitting and generalization issues. We explore hybrid classification methods using the scattering transform, the recursive wavelet decomposition of an input image. We analyse the performance of the scattering transform for the Fanaroff-Riley classification of radio galaxies with respect to CNNs and other machine learning algorithms. We test the robustness of the different classification methods with training data truncation and noise injection, and find that the scattering transform can offer competitive performance with the most accurate CNNs.