Meso-scale structures are network features where nodes with similar properties are grouped together instead of being treated individually. In this work, we provide formal and mathematical definitions of three such structures: assortative communities, disassortative communities and core-periphery. We then leverage these definitions and a Bayesian framework to quantify the presence/absence of each structure in a network. This allows for probabilistic statements about the network structure as well as uncertainty estimates of the group labels and edge probabilities. The method is applied to real-world networks, yielding provocative results about well-known network data sets.