I've written in the past about how many "public" schools are a kind of de facto private property of local homeowners, an idea that Michael Petrilli has given some analytical thought to under the rubric of "private public schools". His definition is public schools "that serve virtually no poor students" and his assessments indicate that there are about 2,800 of them in America.
In a sense that probably understates the scope of the issue. You can imagine an affluent neighborhood where the market price of housing is far too high for low-income families to afford but that also contains a smattering of designated affordable dwelling units as part of an inclusionary zoning scheme. Maybe as a result, it shakes out that five percent of the kids in the local middle are low-income, with 100 percent of those kids occupying the designated affordable units. That's probably good news for those kids. But it still means that the school is in effect closed to newcomers. You've allocated 5 percent of the slots to lucky low-income families, and the rest go to the highest bidder.
To an extent this is an issue cities can solve by not closing tying school assignment to neighborhood. But to an extent the whole process just replicates itself at the district boundary. You often hear for good or for ill some proposal or set of proposals described as a "market-based" reform to the education system. But the fact is that a market-based school choice scheme is at the very core of American public education, it's called the real estate market. That's the appropriate backdrop against which to consider various proposals for changes.