The advent of new sequencing technologies has led to increasing amounts of data being available to perform phylogenetic analyses, with genomic data giving rise to the field of phylogenomics. High-performance computing is becoming an indispensable research tool to fit complex evolutionary models, which take into account specific genomic properties, to large datasets. Here, we perform an extensive Bayesian phylogenetic model selection study, comparing codon and nucleotide substitution models, including codon position partitioning for nucleotide data as well gene-specific substitution models for both data types. For the best fitting partitioned models, we also compare independent partitioning with standard diffuse prior specification to conditional partitioning via hierarchical prior specification. To compare the different models, we use state-of-the-art marginal likelihood estimation techniques, including path sampling and stepping-stone sampling.