First we have to clarify what form of socialism we're talking about. There are many, many different forms, and they will all have different answers; I'll respond from a more mutualist/libertarian socialist perspective.

Second, we have to distinguish that "socialism" does not define a singular state of being. Society is in constant flux with a mishmash of different ideological ideas smashing together and intermingling. Present societies that are generally considered 'capitalist' have a great deal of 'socialist' influences, and we should expect some hypothetical future 'socialist' society to have a great deal of 'capitalist' influence. This is one of the hardest things to describe because it's difficult to say what much of anything would look like under "socialism" because the road to the realization of any broad ideological goal is paved with compromises, shortcomings, and practical failures. So I'll be speaking very, very generally.

Third, we have to talk about what it means to "own" something since "ownership" is heavily tied to housing. Mutualists and others discuss this in terms of property interest rights called usufruct. For mutualists, your right to possess something is determined by how much you use that thing. If you do not actually use something (which is different than profiting from something, to note), you should lose your right to its possession. Generally the boundaries of what "using something" actually means are expected to be defined by some set of norms that may vary by community or be more or less codified into general society, similarly to how we have been taught to intuitively respect the notion of "private property" today (and also its enforcement). Zooming out we also acknowledge that land is held in common, meaning that you cannot simply pick a plot of land that is yours to use without there being common agreement for it to be so -- this is again most likely handled at the community level.

Before you can afford to build or buy a house (and you've moved out from your parents), you would probably live in a form of cooperative housing. This is basically an apartment that you pay to maintain along with the rest of the apartment tenants. The terms of such an arrangement are not absolute and again would likely be highly variable. In general though, you can expect that when you sign a contract to occupy a room, you will pay some amount of money or installment into a mutual fund that handles the expenses of the apartment, and funds that are not used should be returned to you. How the funds are used and the contracts are dictated is either handled by the employees of the apartment firm, you and your fellow tenants, or both (most likely both). The contract may have different terms, such as requiring you to pay out of your own pocket instead of the mutual fund for damages attributable directly to you (like punching a hole in the wall). So you effectively co-own the apartment for as long as you reside there. Alternatively, some communities may confederate housing as a whole rather than just the tenants, since housing is a pretty crucial component of a local economy. This might mean you live in your apartment basically for free (excepting personal damages), but you and the rest of your community pay some general housing tax.

Houses may be handled similarly, or may not. Your possession of a plot of land will have to be vetted in some form by the community. This may be something like a background check, proof of income, survey, interview with the neighbors, etc., processed through some contracted cooperative firm or representatives appointed, elected, or directly managed by the community. The specific procedure is again highly variable and really not that worthwhile to talk about in detail but suffice to say once you have the consensus of the community to possess a specific plot of land, you can either build on it (by contracting some cooperative, yes, paying out of your own pocket) or occupy the existing residence that is there, which you will from then on manage the expenses of until you cease using it.

If you're asking yourself what the point of all these use-defined norms are, it's so that wealth cannot be gained simply through ownership alone. Mutualists believe that you should earn exactly the amount of wealth that you work for in a free market economy -- i.e., you are entitled to the full fruits of your labor and your labor alone.