There are many cases where "property" can be usefully defined, beyond the simple notion of an object or a piece of land being owned.
One example is the radio spectrum. This would be a great candidate. People could buy and sell frequencies and use them for their desired purposes. Spectrum will go to the highest bidder, making for efficient utilization.
It's complicated, because there are a number of technical factors which have to be considered: frequency and bandwidth, power levels, location, modulation and antenna design, etc. Fold all these into an n dimensional parameter space and you can carve up boundaries which identify useful chunks. You have the initial allocation problem, of course, but once you get past that, people can buy and sell, and the market determines the usage.
Today, we mostly handle this by government regulation. This means that frequencies tend to go to those with political pull rather than to the most beneficial uses. A market would be a superior approach.
How would this be handled by those who believe in an absolutist or axiomatic approach to property rights? How can you define what sort of property is appropriate in the context of electromagnetic radiation? I would like to hear what your axioms say about it.
More prosaically, we often divide land-based property by usage. It is common to unbundle mineral rights from surface-dwelling rights. People may also sell off access rights, so that the owner of some other lot has in perpetuity the right to pass over someone else's land in prescribed ways. Once you move beyond the simple concept of land as property, to land-usage as property, you are well into the abstract realm.
I mentioned pollution rights earlier; some areas are experimenting with these. These give polluters the right to generate a certain amount of pollution. They can be bought and sold, and the result is that those businesses for which reducing pollution is most expensive will be the ones which tend to buy the pollution rights, while those which can reduce more cheaply will tend to sell. This automatically limits pollution in the most economical manner possible, a result nearly impossible to achieve with regulation based approaches.
Farmers often have a market in water rights, which allow drawing a certain amount from a water source (river or aquifer). The amount may vary from season to season. The market makes sure that water goes to those crops which provide the greatest overall economic benefit.
Probably the world of finance has produced more forms of abstract property than any other area. Multiple forms of stocks, bonds, options, futures, derivatives, etc., all are claims of various sorts on some underlying property. Now that we have computers, the possibilities are greatly expanding. The result is an increase in the efficiency of the market, which benefits everyone.
We will face new challenges in establishing property rights as we move into space. Already, geosynchronous orbit is a valuable resource. You can only put so many satellites up there without them interfering with each other. A market in satellite positions would be a good way to solve the problem. As with the radio spectrum, there are a number of technical parameters which have to be included in the property bundles: details of the orbital parameters, and possibly also radio frequency limits.
Similar issues will arise with asteroid habitats and free floating space stations. It won't be enough just to have property in the form of land. Access to energy from the sun, orbits which don't come too close, radio allocations, all will need to be parcelled out in order to have a smoothly functioning system. Again, a market is ideal for this purpose, but the property rights involved go far beyond Locke's mixing of labor with land.