Recent observations of neutron stars and nuclear experiments point to substructure in the speed of sound of dense matter, indicating the possible presence of new degrees of freedom. In this talk, I will discuss how this substructure motivates using a flexible equation of state model when inferring the properties of dense matter. Results will include discussion of which physical questions are most naturally answered by model-agnostic analyses of the equation of state. In particular, I will discuss how questions of nuclear physics, such as the existence and character of phase transitions, can be isolated from questions of neutron-star astrophysics, and what can be learned in this regime from existing observations.