Is the preferred basis problem solved?
Unfortunately, physicists and philosophers disagree on what exactly the preferred basis problem is, and what would constitute a solution. Wojciech Zurek was my PhD advisor, and even he and I don't agree. I wish I could give you an objective answer, but the best I can do is state the problem as I see it.
In my opinion, the most general version of the problem was best articulated by Adrian Kent and Fey Dowker near the end their 1996 article "On the Consistent Histories Approach to Quantum Mechanics" in the Journal of Statistical Physics. Unfortunately, this article is long so I will try to quickly summarize the idea.
Kent and Dowker analyzed the question of whether the consistent histories formalism provided a satisfactory and complete account of quantum mechanics (QM). Contrary to what is often said, consistent histories and many-worlds need not be opposing interpretations of quantum mechanics [1]. Instead, consistent histories is a good mathematical framework for rigorously identifying the branch structure of the wavefunction of the universe [2]. Most many-world'ers would agree that unambiguously describing this branch structure would be very nice (although they might disagree on whether this is "necessary" for QM to be a complete theory).
In my opinion, the situation is almost exactly analogous to the question of whether an abstract formulation of classical mechanics (e.g. Lagrangian mechanics) is satisfactory in the absence of a clear link between the mathematical formalism and our experiences. I could write down the math of Lagrangian mechanics very compactly, but it would not feel like a satisfactory theory until I told you how to link it up with your experiences (e.g. this abstract real scalar x = the position coordinate of a baseball) and you could then use it to make predictions. Similarly, a unitarily evolving wavefunction of the universe is not useful for making predictions unless I give you the branch structure which identifies where you are in wavefunction as well as the possible future, measurement-dependent versions of you. I would claim that the Copenhagen cook book for making predictions that is presented in introductory QM books is a correct but incomplete link between the mathematical formalism of QM and our experiences; it only functions correctly when (1) the initial state of our branch and (2) the measurement basis are assumed (rather than derived).
Anyways, Dowker and Kent argue that consistent histories might be capable of giving a satisfactory account of QM if only one could unambiguously identify the set of consistent histories describing the branch structure of our universe [3]. They point out that the method sketched by other consistent historians is often circular: the correct "quasi-classical" branch structure is said to be the one seen by some observer (e.g. the "IGUSes" of Murray Gell-Mann and Jim Hartle), but then the definition of the observer generally assumes a preferred set of quasi-classical variables. They argue that either we need some other principle for selecting quasi-classical variables, or we need some way to define what an observer is without appealing to such variables. Therefore, the problem of identifying the branch structure has not been solved and is still open.
I like to call this "Kent's set-selection problem". I consider it the outstanding question in the foundations of quantum mechanics, and I think of the preferred basis problem as a sort of special case.
The reason I say special case is that the preferred basis problem answers the question: how does the wave function branch when there is a preferred decomposition into system and environment (or into system and measuring apparatus). However, the boundaries of what we intuitively identify as systems (like a baseball) are not always well-defined. (What happens as atoms are removed from the baseball one by one? When does the baseball cease to be a useful system?) In this sense, I say that the decoherence program as led by Zeh, Zurek, and others is an improvement but not a complete solution.
Sorry that's not as clear as you would like, but that's the state of the field as I see it. There's more work to be done!
[1] Of course, some consistent histories make ontological claims about how the histories are "real", where as the many-worlders might say that the wavefunction is more "real". In this sense they are contradictory. Personally, I think this is purely a matter of taste.
[2] Note that although many-worlders may not consider the consistent histories formalism the only way possible to mathematically identify branch structure, I believe most would agree that if, in the future, some branch structure was identified using a completely different formalism, it could be described at least approximately by the consistent histories formalism.
[3] They argue that this set should be exact, rather than approximate, but I think this is actually too demanding and most Everettians would agree. David Wallace articulates this view well.
First some background. The preferred basis problem arises when you consider multiple worlds to be actual different worlds. This is not the same as Everett interpretation of quantum mechanics (coming up soon), it is deWitt's reading of the said interpretation.
The way it works is that an observer picks a number of measurement operators $M_k$ which satisfy $\sum_k M_k^\dagger M_k = \mathbf{1}$. Now the measurement process is no longer a collapse but is instead viewed as an interaction between the observer and the measured system. We imagine that the observer has memory of what was observed, and we shall describe the memory as a quantum state with basis $|0\rangle, |n\rangle$, plus a pre-measurement state $|NA\rangle$. The system starts in some state $|\psi\rangle$ (let's keep it pure for simplicity, but all this works for mixed states too).
Now to the interaction. The joint state of the observer and the system is $|\psi\rangle \otimes |NA\rangle$ which goes to $\sum_k M_k |\psi\rangle \otimes |k\rangle$. This basically means that whenever $M_k$ result has been obtained the observer's memory has recorded the result correctly to have been $k$. This operation is a unitary operation and therefore has an associated Hamiltonian. The job of the observer is to create a device that generates such a Hamiltonian.
So the worlds in this interpretation are entirely observer dependent. The only preferred basis you get is the Schmidt basis that arises from the measurement and the resulting observer being in a mixed state. If the observer and the measured system were well isolated inside some other measurement device then another third observer would actually be able to observe the entanglement between the observer and the measured system and only really see a single world, although for the observer inside there are many worlds (it would be neat to violate Bell inequalities with a physicist and a photon). These worlds arise from the observer's memory.
Finally, you might say that the memory basis of the observer is essentially arbitrary. This is correct. It's arbitrary to the extent that in principle it could be anything. But in practice picking states that are not stable in the world would not produce an evolutionary advantage as they could not be used to faithfully store information. I guess you could say this is where evolution and the interpretation of quantum mechanics meet.