I don't really have a firm stance on most things, and I admit this is an area where I have not really made such a decision. Also, even though I am on the left and socialistic, I'm not that well read about all the details. I just agree with things I read so far, the big pictures and the few details I have read from various sources (cause well, socialism isn't just one thing). I know more about alternative economic models (like RBEs) because I am fascinated by ideas of completely revolutionising our system. *Is an idealist.*
I'm probably just parroting what some others have said, but I dislike the idea of huge expensive property being passed down through generations and people owning far more than they need. I also have noticed it's becoming increasingly harder to purchase a house since they're all being bought by rich people for being let. It's an annoying issue because often letting a house or apartment is quite a bit more expensive (you can buy for around £800 a month but let is £1500!?!?) and you get no return (unlike a bought place, you cannot finish paying it off and live debt free).
Maybe the solution is indeed some kind of lease scheme, as like apartments but shorter amounts of time? But then, who do you pay this money to? Would the government/public own all properties and then the money paid for the lease is basically put back into the public so we can pay for schools and other public services? Hmm, I wonder how this would work.
Obviously, this is thinking in terms of our current system, with currency still being in place and not knowing what the future might hold. There are alternatives but I won't get into those.
Also I think it might be a good idea to clarify what is meant by property in the OP... Although a few posts down has mentioned it. Some people, who are not familiar with these terms, might jump to conclusions of it meaning you don't "own" anything. Why the heck would I want your toothbrush, when I have my own?