We investigate the structure and evolution of multiphase ice-ocean interfaces (‘mushy layers’) and the implications for the geophysics and habitability of ice-ocean worlds. Understanding the potential diversity of these multiphase layers across solar system bodies provides insight into the potential rates and mechanisms of heat and solute transport between their respective oceans and ice shells - which remain largely unconstrained. Additionally, variations in mushy layer properties may drive diverse geophysical processes unique to individual bodies or that may vary regionally on an individual icy world. We explore mushy layer evolution by analytically solving for the thickness of a simplified ice-ocean mushy layer system. We investigate two dynamic regimes, one driven by molecular diffusion and one driven by convection of brine within the mushy layer. We analyze the impact of gravity, thermal gradient, and ocean composition on the thickness of mushy layers. Additionally, a perturbation analysis is carried out to investigate the existence of mushy layer steady states. We show that stable mushy layers exist when ice shells are thickening, suggesting that mushy layers are likely persistent and common features of growing ice shells and accretionary regions of ice-ocean worlds.