The classification of vessel types in SAR imagery is of crucial importance for maritime applications. However, the ability to use real SAR imagery for deep learning classification is limited, due to the general lack of such data and/or the labor-intensive nature of labeling them. Simulating SAR images can overcome these limitations, allowing the generation of an infinite number of datasets. In this contribution, we present a synthetic SAR imagery dataset with ship wakes, which comprises 46,080 images for ten different real vessel models. The variety of simulation parameters includes 16 ship heading directions, 6 ship velocities, 8 wind directions, 2 wind velocities, and 3 incidence angles. In addition, we extensively investigate the classification performance for noise-free, noisy, and denoised ship wake scenes. We utilize the standard AlexNet architecture and employ training from scratch. To achieve the best classification performance, we conduct Bayesian optimization to determine hyperparameters. Results demonstrate that the classifications of vessel types based on their SAR signatures are highly efficient, with maximum accuracies of 96.16%, 92.7%, and 93.59%, when training using noise-free, noisy, and denoised datasets, respectively. Thus, we conclude that the best strategy in practical applications should be to train convolutional neural networks on denoised SAR datasets. The results show that the versatility of the SAR simulator can open up new horizons in the application of machine learning to a variety of SAR platforms.