Reconstructing the evolutionary history of different groups of organisms provides insight into how life originated and diversified on Earth. Phylogenetic trees are commonly used to estimate this evolutionary history. Within Bayesian phylogenetics a major step in estimating a tree is in choosing an appropriate model of character evolution. While the most common character data used is molecular sequence data, morphological data remains an vital source of information. The use of morphological characters allows for the incorporation fossil taxa, and despite advances in molecular sequencing, continues to play a significant role in neontology. Moreover, it is the main data source that allows us to unite extinct and extant taxa directly under the same generating process. We therefore require suitable models of morphological character evolution, the most common being the Mk Lewis model. While it is frequently used in both palaeobiology and neontology, it is not known whether the simple Mk substitution model, or any extensions to it, provide a sufficiently good description of the process of morphological evolution. In this study we investigate the impact of different morphological models on empirical tetrapod data sets. Specifically, we compare unpartitioned Mk models with those where characters are partitioned by the number of observed states, both with and without allowing for rate variation across sites and accounting for ascertainment bias. We show that the choice of substitution model has an impact on both topology and branch lengths, highlighting the importance of model choice. Through simulations, we validate the use of the model adequacy approach, posterior predictive simulations, for choosing an appropriate model. Additionally we compare the performance of model adequacy with Bayesian model selection. We demonstrate how model selection approaches based on marginal likelihoods are not appropriate for choosing between models with partition schemes that vary in character state space (i.e., that vary in Q-matrix state size).Using posterior predictive simulations we found that current variations of the Mk model are often performing adequately in capturing the evolutionary dynamics that generated our data. We do not find any preference for a particular model extension across multiple data sets, indicating that there is no `one size fits all' when it comes to morphological data and that careful consideration should be given to choosing models of discrete character evolution. By using suitable models of character evolution, we can increase our confidence in our phylogenetic estimates, which should in turn allow us to gain more accurate insights into the evolutionary history of both extinct and extant taxa.