It is often claimed that the various interpretations of quantum mechanics are equivalent, in the sense that they yield the same predictions for the outcome of experiments. But for the 3 most popular interpretations, the Copenhagen Interpretation, the Many Worlds Interpretation and the De Broglie–Bohm theory this is only true when making certain (hidden) assumptions, like e.g. that the measurement apparatus is always macroscopic and will thus itself always decohere irreversibly.
An experiment that can distinguish between the Copenhagen Interpretation (CI) and Many World Interpretation (MWI) has been proposed by David Deutsch. Note that falsifying MWI is easy, all you have to do is demonstrate in an experiment that an isolated systems undergoes decoherence that is not due to residual interactions with the environment. This is not as easy as it sounds as any anomalously fast decoherence can be interpreted as an interaction with new physical degrees of freedom (e.g. dark matter particles, interactions via extra dimensions etc. etc.) that then need to be tested for in different experiments.
But how can we experimentally falsify CI? David Deutsch has proposed the following (thought) experiment. Suppose that large scale quantum computing makes it possible to implement an AI in a quantum computer. You could even imagine uploading your brain continents to a quantum computer. Then suppose that you are alive inside a virtual quantum simulation where you set out to measure the z-component of a spin (that is then implemented by a qubit) that is polarized in the x-direction. Then consider a transform to the state this is the initial state except that you keep a record in your memory that a measurement was actually performed. It's easy to see that this transform is unitary, and it is known that a quantum computer can approximate any unitary transform to arbitrary accuracy.
So, this means that you can undo the measurement, while you can still verify that a measurement was actually carried out (the result of the measurement will necessarily be erased). The spin will be restored in the original state, and you can verify by measuring the spin in the x-direction, and repeating this whole experiment many times and verifying that you always find the same spin.
Now, if only one branch of the wavefunction is physically realized, then the unitary transform back to the original state except for keeping the information about the measurement being carried out, would not work. If you apply the unitary transform to only one branch, the state of the spin won't be restored to the initial state. So, this experiment would verify to you that when you measured the z-component of the spin that both possible outcomes were indeed physically realized.
De Broglie–Bohm theory is also said to be equivalent to Copenhagen Interpretation, but just like the case of the MWI, this isn't true either. De Broglie–Bohm theory is only equivalent to regular QM if one assumes quantum equilibrium, deviations from this state can be tested as it yields probabilities that deviate from the Born rule.
So, the answer to the question "by what logic does one come to support a particular interpretation of quantum mechanics?" is just as anything else in physics, the outcome of experiments. The only problem is that we don't have the results to rule out any of the existing interpretations, but that doesn't necessarily mean that they are all equally plausible given what we do know. One can consider thought experiments that will yield different outcomes depending on your interpretation, or you may interpret what is going on in some particular thought experiment differently depending on what your interpretation of QM is. Such arguments are valid arguments in physics.