Canada

(Back-of-the-napkin ideas from a nerd who has way to much time on their hands to read about stuff like this)

Does everyone live in social housing or is there just a lot of social housing with a secondary open market?

likely the second: it's human nature to have preferences for things, and if you don't accommodate for people wanting something outside the norm, it'll just end up on a gray or black market. That's not a defeatist "what's the point in even trying" statement ("why ban xyz?? People will just get xyz anyways!" is not what I'm trying to say), just acknowledging the realities of people being an extremely diverse collection of individuals and how to work that fact into on-the-ground policy. If you're in a position to be able to afford something else, you should be able to build it - so long as the external costs such as utility hookups and roads and so on are factored into "can afford" (property taxes should be replaced with a land value tax, or in lieu of that raised to cover the costs of infra to get to your home).

Are there restrictions on investing in housing, such as you may only own 1 home or maybe 2?

This would help slow the bleeding, but I don't see it being a solution: numbered corps "suddenly" on the rise after that happens. The only real solution is for the government to build a ton of social housing themselves and not P3 it or leave it to the market, since we're trying to leave it to the market and it's obviously failing to deliver and P3s are a waste of money.