School choice can segregate schools by academic ability, income or ethnicity, but is this because of households' choices, or constraints in access to good schools? We examine whether segregation is by choice, finding that households' school choices are segregating in most areas. Through counterfactual simulation, we find that implementing a policy of "neighbourhood" schools would, in contrast, reduce segregation in most areas, under the assumption that each household's location is fixed. Policymakers require further evidence to weigh up the effects of school choice systems on sorting across schools and neighbourhoods, relative to potential efficiency benefits of school choice.