Although Mars does not possess a magnetic field today, parts of its crust are strongly magnetized, suggesting a limited early dynamo, likely powered by rapid heat flow from the core. If the core is undergoing crystallization, the associated compositional changes would provide an additional mechanism for driving convection—possibly the dominant driver for the Earth’s dynamo today. This raises the question: does the lack of a global dynamo field on Mars suggest the absence of a partially crystallized core? More generally, what is the range of possibilities for the history and future of the Martian dynamo and which scenarios would be ruled out by the presence or absence of a solid inner core? Here we develop a new internal structure, thermal evolution, and buoyancy flux model to investigate the conditions under which the Martian core could experience compositionally-driven convection, either in the past or the future. We show that the presence of a partially crystallized core is compatible with the lack of a dynamo today but that such a scenario implies the Martian dynamo could reactivate at some point in the future. We find that convection driven by top down core crystallization (iron snow) can occur only when light elements do not partition strongly into the liquid versus the solid phase. Our model identifies the key variables for determining which dynamo regimes are possible, can help in assessing implications of future observations relating to the Martian core, and forms the basis for further comparative study across rocky planets.