From a recent paper that is getting non-trivial attention...

"Quantum states are the key mathematical objects in quantum theory. It is therefore surprising that physicists have been unable to agree on what a quantum state represents. There are at least two opposing schools of thought, each almost as old as quantum theory itself. One is that a pure state is a physical property of system, much like position and momentum in classical mechanics. Another is that even a pure state has only a statistical significance, akin to a probability distribution in statistical mechanics. Here we show that, given only very mild assumptions, the statistical interpretation of the quantum state is inconsistent with the predictions of quantum theory. This result holds even in the presence of small amounts of experimental noise, and is therefore amenable to experimental test using present or near-future technology. If the predictions of quantum theory are confirmed, such a test would show that distinct quantum states must correspond to physically distinct states of reality."

From my understanding, the result works by showing how, if a quantum state is determined only statistically by some true physical state of the universe, then it is possible for us to construct clever quantum measurements that put statistical probability on outcomes for which there is literally zero quantum amplitude, which is a contradiction of Born's rule. The assumptions required are very mild, and if this is confirmed in experiment it would give a lot of justification for a phyicalist / realist interpretation of the Many Worlds point of view.

More from the paper:

"We conclude by outlining some consequences of the result. First, one motivation for the statistical view is the obvious parallel between the quantum process of instantaneous wave function collapse, and the (entirely non-mysterious) classical procedure of updating a probability distribution when new information is acquired. If the quantum state is a physical property of a system -- as it must be if one accepts the assumptions above -- then the quantum collapse must correspond to a real physical process. This is especially mysterious when two entangled systems are at separate locations, and measurement of one leads to an instantaneous collapse of the quantum state of the other.

In some versions of quantum theory, on the other hand, there is no collapse of the quantum state. In this case, after a measurement takes place, the joint quantum state of the system and measuring apparatus will contain a component corresponding to each possible macroscopic measurement outcome. This is unproblematic if the quantum state merely reflects a lack of information about which outcome occurred. But if the quantum state is a physical property of the system and apparatus, it is hard to avoid the conclusion that each marcoscopically different component has a direct counterpart in reality.

On a related, but more abstract note, the quantum state has the striking property of being an exponentially complicated object. Specifically, the number of real parameters needed to specify a quantum state is exponential in the number of systems n. This has a consequence for classical simulation of quantum systems. If a simulation is constrained by our assumptions -- that is, if it must store in memory a state for a quantum system, with independent preparations assigned uncorrelated states -- then it will need an amount of memory which is exponential in the number of quantum systems.

For these reasons and others, many will continue to hold that the quantum state is not a real object. We have shown that this is only possible if one or more of the assumptions above is dropped. More radical approaches[14] are careful to avoid associating quantum systems with any physical properties at all. The alternative is to seek physically well motivated reasons why the other two assumptions might fail."

On a related note, in one of David Deutsch's original arguments for why Many Worlds was straightforwardly obvious from quantum theory, he mentions Shor's quantum factoring algorithm. Essentially he asks any opponent of Many Worlds to give a real account, not just a parochial calculational account, of why the algorithm works when it is using exponentially more resources than could possibly be classically available to it. The way he put it was: "where was the number factored?"

I was never convinced that regular quantum computation could really be used to convince someone of Many Worlds who did not already believe it, except possibly for bounded-error quantum computation where one must accept the fact that there are different worlds to find one's self in after the computation, namely some of the worlds where the computation had an error due to the algorithm itself (or else one must explain the measurement problem in some different way as per usual). But I think that in light of the paper mentioned above, Deutsch's "where was the number factored" argument may deserve more credence.

Added: Scott Aaronson discusses the paper here (the comments are also interesting).