Disagreements persist on how to design a self-sufficient, carbon-neutral European energy system. To explore the diversity of design options, we develop a high-resolution model of the entire European energy system and produce 441 technically feasible system designs that are within 10% of the optimal economic cost. We show that a wide range of systems based on renewable energy are feasible, with no need to import energy from outside Europe. Model solutions reveal considerable flexibility in the choice and geographical distribution of new infrastructure across the continent. Balanced renewable energy supply can be achieved either with or without mechanisms such as biofuel use, curtailment, and expansion of the electricity network. Trade-offs emerge once specific preferences are imposed. Low biofuel use, for example, requires heat electrification and controlled vehicle charging. This exploration of the impact of preferences on system design options is vital to inform urgent, politically difficult decisions for eliminating fossil fuel imports and achieving European carbon neutrality.