I think this has been much relaxed recently. You just need to prove that you live in Shanghai for more than 6 months (rental is okay) and either pay social security taxes or are registered as seeking employment. Thanks to the one child policy, if they don't relax enrollment policy the schools are going to run out of students.
No, not just that, there are now specific rules to place those without hukou ahead of those who have.
e.g. in my area, say kid's dad has hukou in the local area and kid's mum has hukou from another area in Shanghai, then the kid will get lower priority for applying schools in the area comparing to those with one of the parents with hukou in the area and another one without hukou in Shanghai at all. The logic is that the first couple has the option to send their child to a different area in Shanghai while the second one doesn't have such option available to them.
I think that only applies to middle class chinese with formal resident permits. Migrant workers from the villages, which make up much of that 12 million, still have to leave their kids at home. Anyways, shanghai is a bad example of equitable education.
School enrollment is permitted for those with temporary resident registration (which most everyone should qualify) but requires three year continuous registration, so is still a high bar to cross. Hope to see continued relaxation. They will need the younger generation to fill the schools as fertility rate stays low.
The condition for enrolling in public schools: http://www.shanghai.gov.cn/nw2/nw2314/nw2319/nw12344/u26aw55...
The condition for obtaining residency registration: http://zwdt.sh.gov.cn/zwdtSW/zwdtSW/juzhuzheng/lsjzz.jsp