I've seen the preferred basis problem referred to in many places, but have not seen a clear explanation of what the problem is. For example, this question asks whether the problem has been solved, but I wasn't able to glean anything from the discussion on what the problem was. Zurek 2001 gives the following:
the original MWI does not address the "preferred basis question" posed by Einstein
with a footnote quoting Einstein as saying this:
When the system is a macrosystem and when ψ1 and ψ2 are 'narrow' with respect to the macrocoordinates, then in by far the greater number of cases this is no longer true for ψ = ψ1 + ψ2. Narrowness with respect to macrocoordinates is not only independent of the principles of quantum mechanics, but, moreover, incompatible with them.
The first sentence seems obvious to me. I'm baffled by the second sentence. Some states are narrow and some are not. What's the problem, and what does this have to do with a basis or a preferred basis?
Zurek, "Decoherence, einselection, and the quantum origins of the classical," http://arxiv.org/abs/quant-ph/0105127