In the name "interpretations" it is implied that these are believed to not be empirically distinguishable. What everyone agrees on is how to calculate empirically testable predictions. That is essentially mathematics, and if you are very mathematically minded, you can propose to axiomize it, that is, find a minimum number of axioms (postulates) from which it follows. Obviously, as with any system of axioms, there is a bit of personal preference involved: Many call the time-dependent Schroedinger equation such a postulate. I personally prefer to only use de Broglie's matter-wave duality as a postulate and derive e.g. the (non-relativistic) Schroedinger equation for free massive particles from the non-relativistic impulse-energy relationship expressed for plane waves (which are one possible and hence sufficient choice for a basis system to express any wave in). I am afraid all one can summarily say is that there is no widespread consensus on what postulates/axioms to use; we physicists only agree on the (well-tested) outcome and perhaps that how exactly one gets there is at least as much a question of mathematics as of physics.
Whilst "interpretations" was certainly the correct word historically, one can make good arguments that some theories that carry the historical name "interpretation" can actually be tested, at least if you take a sufficiently broad approach to tests as to permit thought experiments. Most interpretations fail to resolve paradoxes, such as Schroedinger's cat, the EPR paradox, or how to determine what constitutes a measurement (or how a wave-function collapse is physically possible considering quantum mechanical evolution only permits unitary, that is non-collapsing, time-evolution). That means most "interpretations" do not pass the test of logical consistency since otherwise there would not be any such paradoxes.
Even with regard to experimental tests, we have at least two: The violation of the Bell inequality has been experimentally demonstrated, which is a test against hidden local variables, which would otherwise remain fair game for outlandish "interpretations." And a simplistic form of Schroedingers cat can be realized in a 2-qubit quantum computer, where modelling decoherence as interaction with an environment (that could be a real uncontrolled environment or simulated via more qubits) provides an experimentally verifiable detail theory of just how the cat would decohere out of its curious superposition. If you like, that constitutes an empirical test that the Schroedinger cat paradox is not really a paradox and any interpretation that is unable to resolve it must be an incorrect or at least incomplete theory.
Finally there is the issue of the recurrence time. Since quantum mechanics only allows unitary transforms, essentially rotations (and reflections) in a high dimensional Hilbert space, it predicts that everything repeats eventually, although most likely, due to the high dimensionality, only after a time that is mind-bogglingly huge even when compared to cosmological timescales. That is at odds with thermodynamics and relativity (at least if that is valid for ever expanding universes). There obviously must be tests for it, but finding them is surprisingly difficult. For example, even in theory waiting out the recurrence time is frustrated by the fact that if it exists as such, all notes and memories would have reverted, and we would again wonder if we should start the experiment for the first time ever! Yet on systems small enough that we can isolate them sufficiently from the environment to have short and observable recurrences (that experimentalists have come to call "revivals"), they demonstrably occur. At least to the extent that you consider this good enough to be at least a partial test of recurrence for the universe as a whole, this is of course at odds with any postulate of wavefunction collapse.