Many of the differences between the quantum and classical worlds are most visible when studying measurements. For example, classical measurements are repeatable and their outcomes are objective, whereas only measurements of quantum systems which exhibit quantum Darwinism have these properties and so support the emergence of classicality.  Another difference is that the outcomes of sets of classical measurements can always be described with joint probability distributions, whereas sets of quantum measurements may require quasiprobabilities. Only some quantum models appear classical in the sense that they only require ordinary probabilities to describe joint measurements. In this talk, I will take these two notions of how a quantum model may appear effectively classical and provide a broad classification of Hamiltonians which support them. Interestingly, the two approaches each lead to exactly the same set of Hamiltonians, implying that these properties of classical measurements are deeply related.