The vast majority of Singaporeans live in apartments they own, and don't pay rent. However you are right that most of these apartments were built by an arm of the government, see https://en.wikipedia.org/wiki/Housing_and_Development_Board
There are grants for lower income people to make it easier for them to buy a home. Some people also rent directly from the government, but that's the exception. Most own.
Housing ain't cheap in Singapore. Whether you measure that in terms of rent, or in terms of monthly mortgage costs, or in terms of the opportunity cost of capital (for those who own their homes outright). As everywhere else in the world that's mostly a function of supply and demand, and where that supply comes from (public, private, etc) doesn't really matter too much.
Singapore has been building a lot of housing, and is still building a lot of housing. Both by public and private developers. But we are living on a small island with lots of people, and thanks mostly to immigration our population is still growing. (I myself am an immigrant here.)