I can identify four main ones.
Title based (known as "private property" today): Title to a property can be purchased, traded, or aquired, giving the title holder certain exclusive rights of control over the property, enforced by the state, community, or private enforcement.
Use/maintenance based (sometimes known as "usurfruct"): The right of control is enforced based on who is considered to "use" and/or "maintain" the property, enforced by state, community, or private enforcement. Through use and upkeep they are considered the owners of the property and as such their rights to the property are defended when they are not present.
Occupancy based: The state, community, or private enforcement defend the authority of whoever is currently occupying a property, until the moment they leave said property of their own volition. No absentee rights.
Centralized: Ownership/control of all properties is administered centrally by the state or community. Instead of enforcing individual rights, the enforcement bodies take over management.
Now, title based is obviously private property as we know it, and could be ramped up to greater degrees for anarcho-capitalism. Use/maintenance is what some people who call themselves "free market socialists" want, and an occupancy based system with no absentee rights whatsoever is what farther left anarchists than that want; a system in which people can only occupy or possess property and there are no absentee rights. To a limited degree it represents squatter's rights today.
Pure centralized "ownership" can be spun either as the state socialist ownership system, or some sort of state capitalism, purely depending on how "proletarian" you decide the centralized administration is.
The societies we live in now are largely a combination of centralized property and title property with the one being made less pure by the other, and so we have the "mixed economy".
Looked at this way, and getting past the tired argument over "socialism" or "capitalism". I would personally argue for less centralized administration generally. I also think that we need to balance use/maintenance rights against title based rights.
For example, when it comes to the most basic things like having a place to live we should emphasize the enforcement of use rights over title. Therefore, we should transfer rented accommodation over to use title to prevent homelessness.
Looking at property issues in this way, we can individually analyze elements of the system and what type of property should be emphasized without getting into the by now meaningless question of whether we should have a "socialist" or a "capitalist" system overall.