We address the problem of how to use a sample of trajectories to choose from a candidate set of possible state spaces in different types of Markov processes. Standard approaches to solving this problem for static models use penalized maximum likelihood criteria that take the likelihood of the trajectory into account. Surprisingly, these criteria do not work even for simple fully observable finite Markov processes. We propose an alternative criterion and show that it is consistent. We then provide a guarantee on its performance with finite samples and illustrate its accuracy using simulated data and real-world data. We finally address the question of model selection in Markov decision processes, where the decision maker can actively select actions to assist in model selection.