There is much activity and debate on these interpretations. Part of it is related to the measurement problem. Is there such a thing as quantum collapse? How does it work?
David Mermin once said in an article in Physics Today that new interpretations are added every year and none has ever been ruled out. If this is true, then it indicates that the interpretations of quantum mechanics is not part of science, and therefore also not part of physics.
I am not going to say one should not work on such interpretations and try to make sense of what is going on, but the scientific method does not seem to help us here. Perhaps people will eventually come up with experiments to determine how nature works. I’ve seen some proposals, but they are usually associated with some new mechanisms, which in my view are unlikely to be correct.
It occurs to me that while we cannot say which of the interpretations are correct, we may just as well just pick one and work with that. So I pick the simplest one and when I want to figure out how things will work out in one of these experiments, then I can just consider how things will work according to this interpretation. If such a prediction turns out to be wrong, it would show that this interpretation (and all those that made the same prediction) is wrong after all.
The simplest interpretation according to me is the many-world interpretation. It is simple because it does not require the weird unexplained notion of quantum collapse. People don’t like it, because it seems to require such a lot of different worlds. For that reason it is also associated with the idea of a multiverse.
Well no, those ideas are anyway misleading. In quantum mechanics, all interactions are described by unitary evolution. The picture that it represents is that there is a set of states that the universe can take on. One can think of each such state as a different description of the world. Hence “many worlds.” However, the actual state of the universe is a quantum superposition of all the possible worlds. In the superposition each world is associated with a complex probability amplitude. It means that some worlds are more likely than others. During interactions these probability amplitudes change.
That is the whole idea of unitary evolution. All the possibilities are already present right from the start. The only thing that interactions do is to change the probability amplitudes that are associated with the different worlds. During the evolution in time the different worlds in the superposition can experience constructive or destructive interference, which would change their probability amplitudes, making some less or more likely that they were before.
The number of worlds (number of terms in the superposition) stays the same. They don’t increase as a result of interactions. How many such worlds are there? Well, if we look at the properties of the set of such basis states, then it is often assumed to be a countable infinite number. However, it may turn out to be uncountably infinite, having what is called the cardinality of the continuum.
What is more is that these different worlds are not distinct unique worlds. One can redefine the basis set of worlds by forming different superpositions of the worlds in the original set to get a new set in which the worlds now look different.
How does all this relate to what we see? The dynamics of the universe causes the interferences due to the unitary evolution to favor a small set of worlds that look very similar. This coherence in what the world looks like is a result of the constructive interference produced by the dynamics.
So the world that we see at a macroscopic level is not just one of these worlds. It is, in a sense, a conglomeration of all those worlds with large probability amplitudes. However, the differences among all these worlds are so small that we cannot notice it at a macroscopic level.
OK, not everything I said here can be confirmed in a scientific way. I cannot even proof that the many-worlds interpretation is correct. However, by thinking of it in this way, one can at least get some idea of it that makes sense.