What happened to the property (tanks, submarines, even just tractors) of the soviet union?

There's a couple things to be disentangled here.

First is personal property in the Soviet Union. While it's true that the varieties of personal property weren't as broad as in Western countries (no stocks, no commercial property in private hands, and an extremely restricted amount of real estate), it did still exist: Soviet citizens could hold bank accounts, own personal vehicles and even in very specific circumstances build and own country houses, as I discuss a little here.

Now, I by no means want to be an expert on how titles worked in Soviet law - maybe u/KongChristianV could help there, but while there could be a lot of regulatory change between aspects of ownership (especially of those private dacha cottages), this personal ownership was never really questioned in a legal sense. I say in a legal sense because sure, someone with a gun could absolutely pull you over in the 1990s and say the car is his, and maybe even have some police officers or even a judge say "oh yeah, that's his now", but everyone would clearly know that what was really happening was straight-up theft protected by corruption.

It's worth noting a few other aspects of property in the USSR. One is that most people by 1991 were living in apartments, and these apartments werr municipally-owned. They were privatized in the 1990s, but this process involved basically creating all the aspects and institutions of a real estate market from scratch.

Another is that while most non-personal property was owned by the state (whether that's federal, republican, provincial or municipal could vary), there were exceptions.

One exception was cooperative property. While state farms became fashionable in the later Soviet period, the original collective farm (and many still existed in the later years) was a version of a coop: in theory members were contributing resources to a collective endeavor, and received a calculated payout of the annual surplus (often this was very meager). Collective farmers (again, depending on the year and regulations) were usually able to own some private livestock and grow private crops in "kitchen gardens", and much of this could even be sold privately in markets. A new Law on Cooperatives took effect in 1988, which allowed other sorts of business coops to form, the idea being that they would be groups of workers who were providing some sort of contract services to state-owned enterprises (the reality is that these were often means for coop partners and enterprise managers to siphon off hard currency from state owned enterprises, but that's another story).

Another big exception to state-owned property was Communist Party property, which was in many ways a state-within-a-state, but still technically separate. Much of the perks party officials enjoyed was technically access to these properties, from dachas to special food deliveries and even furniture, with the unspoken understanding that what the party provided could be taken away.

Anyway, what ultimately happened was this. The republics' governments, starting in 1990, asserted "sovereignty", claiming primacy of republican law over union law and claiming ownership of resources within republic borders. This was initially a political ploy and bargaining position with regards to the Union government, but after the failure of the August 1991 coup, it rapidly took on real meaning. A big blow to union-level institutional ownership was when Boris Yeltsin banned the Communist Party on Russian soil and seized its property - this made the brand-new Russian presidency a massive property owner overnight. I describe more of that process here. A similar process happened to Union government properties and institutions over the rest of the year.

Ultimately, this meant that Russia and the other former Soviet Republics had governments owning most non-personal former Soviet property at their disposal, and vast quantities of this were privatized in the 1990s. Much of these privatizations were incredibly controversial (such as the 1992 plan where Russian citizens were issued coupons for ownership of state enterprises, but almost all of these ended up in the hands of oligarchs, and the even more controversial 1995 Loans for Shares scheme). I discuss some of those processes here.