They are publicly built, but people can (and do) buy them from the state. I was living in a public housing project from the 40s (Tiong Bahru) when I was living there. It was one of the earliest such projects, and is now a conservation area. In the current market, my old flat with 4 rooms, two bathrooms, 1 kitchen would net around 1.2 million SGD. I was renting from the private owner for 4k SGD a month.