What constitutes an interpretation of a scientific theory? Are interpretations subject to scientific criteria like falsifiability, repeatability, etc, or are they philosophical comments? And finally, are they necessary/useful?
This question is motivated by another question: Is the Copenhagen interpretation falsifiable?
Aside from minimal interpretations, which simply relate theoretical abstractions to empirical/practical procedures, interpretations are typically treated as philosophy/metaphysics. As such they can be used as vague blueprints for extending/modifying the theory, this was Popper's own view, he called them "metaphysical research programs". On more skeptical views of theory/observation dichotomy (Quine, Kuhn, Lakatos) the distinction between formalism and interpretation can not be made at all, it amounts to purely pragmatic loose talk, and it is scientific systems or "paradigms", with mixed formal and interpretational elements, that face "tribunal of experience" as wholes, see Theory and Observation in Science. But even more conservative philosophers of science largely no longer attempt to offer "strict" criteria, like falsifiability, verifiability, etc., for distinguishing "scientific" claims. Given the variation in scientific approaches from history to physics formulating any general, but still meaningful, scientific method is seen as largely hopeless. "Testability" is a loose term used to replace these older criteria, but determining its content is largely left to practice in specific sciences.
The role of "metaphysics" in developing new theories is recognized by physicists themselves, Feynman even addressed it in his Nobel lecture. As he points out, while different interpretations of the same formalism may not be testable on its existing evidentiary base, they, or rather extensions of the formalism developed from them, may become testable in the future. And this certainly makes wide variety of interpretations useful:
"Theories of the known, which are described by different physical ideas may be equivalent in all their predictions and are hence scientifically indistinguishable. However, they are not psychologically identical when trying to move from that base into the unknown. For different views suggest different kinds of modifications which might be made and hence are not equivalent in the hypotheses one generates from them in ones attempt to understand what is not yet understood. I, therefore, think that a good theoretical physicist today might find it useful to have a wide range of physical viewpoints and mathematical expressions of the same theory (for example, of quantum electrodynamics) available to him.
This may be asking too much of one man. Then new students should as a class have this. If every individual student follows the same current fashion in expressing and thinking about electrodynamics or field theory, then the variety of hypotheses being generated to understand strong interactions, say, is limited. Perhaps rightly so, for possibly the chance is high that the truth lies in the fashionable direction. But, on the off-chance that it is in another direction - a direction obvious from an unfashionable view of field theory - who will find it? Only someone who has sacrificed himself by teaching himself quantum electrodynamics from a peculiar and unusual point of view; one that he may have to invent for himself. I say sacrificed himself because he most likely will get nothing from it, because the truth may lie in another direction, perhaps even the fashionable one."
As for the Copenhagen interpretation, its testability is more problematic than usual because it is largely a vague catch-all with no particular content, vague even by the low standards of interpretations in general, see What is the philosophy behind the Copenhagen interpretation of quantum mechanics? For one, Bohr's original, Heisenberg's, and later textbooks' versions of Copenhagen are three different things. But alternative Everett's, Bohm's, and especially objective collapse, interpretations do suggest extensions of quantum theory that can be tested. So if they are confirmed Copenhagen would presumably be disconfirmed.
An interpretation of a theory, in particular quantum theory, is said to be an account of what that theory is saying about the world that can be cleanly separated from the theory's predictions. People who accept this idea claim that different interpretations of quantum theory can't be tested, and the question of which interpretation is true is not solvable.
This line of argument is incorrect. One problem with this idea is that predictions of a theory are accounts of what is happening in the world according to some theory under some particular set of circumstances. Without such an account there is no such thing as whether an experiment has been set up properly, and so an experiment can't be set up. So an account of what the theory claims about reality can't be separated from its predictions, and this is just as true of quantum theory as of any other theory. Another problem is that scientific theories in general are not about predictions, they are about what is happening in reality. The predictions are a means of testing an account of what is happening in reality, they are not the whole point of having a theory. For example, lots of dinosaurs were never fossilised. The existence of those dinosaurs doesn't lead to any particular prediction, but it is a consequence of the theory that dinosaurs existed and of the mechanisms of fossilisation. You can't separate that consequence from the rest of the theory and ignore it without ruining the explanation that the dinosaur theory provides.
Various theories are said to be interpretations of quantum mechanics. Some of these, such as the pilot wave theory, have a different account of what is happening in reality from quantum theory and try to make that account produce testable predictions different from quantum mechanics. Some intepretations, such as the Copenhagen interpretation, say stuff that is vague or inconsistent, e.g. - sometimes quantum theory describes reality and sometimes it doesn't. Such theories fail to make predictions because they fail to give an account of what is happening in experiments. There is one interpretation that takes quantum theory seriously as a description of reality - the many worlds interpretation. The MWI makes testable and specific predictions and gives an account of what is happening in reality - reality consists of a structure that in some circumstances looks like a collection of universes as described by classical physics, see
https://arxiv.org/abs/quant-ph/0104033
and "The Beginning of Infinity" Chapter 11 by David Deutsch.
