As someone else said, there is La Maison du Père that provide (almost) free shelter.
Otherwise, there are provincial, municipal and private orgasms that help as they can with some services for reinsertion. Like the “L’Itinéraire” magazine.
The SPVM (police department) are also there to help during interventions with people with mental illness, in crisis, or to give references for some government’s services. During great cold they are often outside to distribute goods and coffee. They don’t just give fines.
You might not be aware of that, but there is what we call a “housing crisis” right now. There is not enough place to house everyone, and there are not enough construction sites to fix the problem rapidly considering the recent increase in population. It will take years to adjust. You can’t just make a bunch of apartments appear out of nowhere. Doesn’t matter the policy you apply to distribute them.
And about social housing, yeah, everybody likes that. It’s more a matter of government inefficiency rather than a lack of will from the population.