In this paper, we assess the suitability of a number of different machine learning (ML) methods for detecting shock fronts in Computational Fluid Mechanics (CFD) simulations. Detection and handling of shock fronts is important in a wide variety of fluid mechanics problems. We focus on computational astrophysics, where a successful algorithm must be able to classify the simulated fluid elements as belonging to a no-shock, ahead-of-the-shock-front and behind-the-shock-front class. We implement and test several supervised multi-class classification ML methods for highly imbalanced classes. The training data set is generated by an exact solver for a Riemann Problem [1] (one of the most straightforward non-trivial CFD tests). The most suited algorithm(s) are chosen according to their accuracy, speed, and ease of training. Our preliminary results show that the random forest algorithm (with class balancing) is the best method for classification.