Summary: Two approaches for model-based clustering of categorical time series based on time-homogeneous first-order Markov chains are discussed. For Markov chain clustering the individual transition probabilities are fixed to a group-specific transition matrix. In a new approach called Dirichlet multinomial clustering the rows of the individual transition matrices deviate from the group mean and follow a Dirichlet distribution with unknown group-specific hyperparameters. Estimation is carried out through Markov chain Monte Carlo. Various well-known clustering criteria are applied to select the number of groups. An application to a panel of Austrian wage mobility data leads to an interesting segmentation of the Austrian labor market.