Is the measurement problem an interpretation or practical problem? According to Wikipedia:

In quantum mechanics, the measurement problem is the problem of how, or whether, wave function collapse occurs.

Is the measurement problem an interpretation problem or a practical problem?
If the measurement problem is an interpretation problem, then there is no experiment that we can perform which can give a definitive answer to the measurement problem. The problem is then not what happens, but how we interpret it. So, if we perform ‘action X’, then we know exactly what we will observe (with some probability), but the measurement problem is what this means.
If the measurement problem is a practical problem, then we can find answers using physical experiments. For example, we could test whether ‘action X’ causes ‘the wavefunction to collapse’, which can be seen from ‘result Y’. Maybe the experiment is outside our current abilities; but in principle the problem can be tested.
 A: It is neither an interpretation nor a practical problem - it is a an open question in the theory, a place where the theory (that is, quantum mechanics) is not complete.
QM says the quantum state of a system evolves unitarily. It also says that, after a measurement has been performed then we cannot consider the quantum state that allowed us to model the system so far as valid anymore. Instead the system must be represented by a new quantum state, that takes into account the outcome of the measurement. And that is not an interpretation, it is plain QM.
Now what count as a measurement? What are the criteria allowing to make the theoretical difference, in the model, between an interaction and a measurement? QM does not provide the answer to this question, so in that regard it is incomplete, and that is the measurement problem.
A: As far as we know, the measurement problem is an interpretative problem. We know of no experiment that can tell how or whether the wave function collapses, or even whether the wave function is part of objective reality or simply a useful model.
The instrumentalist or "shut up and calculate" school of thought in physics holds that even thinking about the measurement problem is a waste of time, since it is (according to instrumentalism) fundamentally undecidable and has no impact on the outcomes of experiments.
A: It's...not that simple.
Some people believe that operational quantum mechanics is a complete theory: The formalism of quantum mechanics makes probabilistic predictions for all measurements, that's all a physical theory can ever hope to achieve, and the discussion around the measurement "problem" is pure metaphysics with no practical relevance - nothing the interpretations do can improve what kinds of predictions we make.
Some people believe that a fundamental theory of nature shouldn't only make probabilistic predictions, i.e. that the "real" theory must be a (super)deterministic hidden variable theory, and that these hidden variables must be, in principle, knowable. Of course, if there is a deterministic theory of nature and its parameters are in principle knowable, then operational quantum mechanics is not the best theory of nature we can have: The deterministic predictions from this hidden variable theory would be an improvement over the probabilistic predictions of quantum mechanics.
This is largely why Bell's theorem - and other no go theorems concerning hidden variable theories - are of such great foundational interest: The more kinds of hidden variable theories are excluded from being compatible with our observations, the less probable this stance becomes. Conversely, as long as deterministic theories are not fully excluded, it is still legitimate to doubt the operational view that the probabilism of current quantum mechanics is as good as it gets.
So, while you can believe in hidden variable theories whose parameters are in principle unknowable (e.g. someone who believes in Bohmian mechanics and thinks we cannot ever in principle resolve the initial conditions of a particle so well that we could actually make deterministic predictions), and have no issue with claims that quantum interpretations are metaphysics/not a "practical problem", some interpretations (or rather their adherents) have a definite interest in labeling the measurement problem as "just about interpretations" and others have a definite interest in claiming they have practical relevance, and one way this split aligns is along the question of whether or not the respective interpretation believes in hidden variables: Saying the measurement problem is not about differences in predictions - i.e. not a "practical problem" - is saying that you believe that fundamentally there cannot be an underlying deterministic hidden variable theory whose parameters are knowable to a degree where we could make deterministic predictions, so it is already some kind of interpretational statement if we're being precise!
This isn't the only aspect, there's other philosophical reasons due to which people can disagree over whether or not quantum interpretations are a "practical problem": For instance, while classic Popperian falsificationists should have no fundamental problem with leaving quantum interpretations as metaphysics, there are others - often some kind of positivist - who find it necessary for a physical theory to ascribe a definite ontology to the world, explaining how the world "really is". These positivists cannot agree to treating quantum interpretations as a minor or interpretational issue because the interpretations are exactly about the ontological questions that, to them, are an essential part of a physical theory.
