David Deutsch in The Fabric of the Universe says that Quantum phenomena are actually the interference of the the different universes in the multiverse. They have written programs that if they run on quantum computers and give answers then it that will show that the multiple universes exist because the only way to do the computation is by doing it in multiple universes at the same time other wise it would take too long to compute. So this is a specific experiment that is planned when a Quantum computer exists.
The most convincing argument is that the Multiple Worlds theory of Hugh Everett is that it is the simplest theory, with the fewest assumptions. But it has two problems, one is that we cannot see the other universes, and the other is that it is assumed that all possible paths must be taken and so there is an explosion of universes at each quantum observation or event that can have multiple outcomes.
I propose that another way to think of this is that there is in fact a finite number of universes and that they are related to each other in by what I call an Emergent Meta-system. An Emergent Meta-system is sort of like a Genetic Algorithm which is creating and destroying different universes as Quantum observations are made and multiple paths are possible. We are in the universe that randomly we observe and the other universes are like ghosts that are possibilities that we could have been in if the random by observations had turned out otherwise. Modal logic is still not understood very well and the exact relation between actual and possible is still a mystery. I like the formulation of Ian Thompson based on propensities at http://www.ianthompson.org/.
The major problem is that we can only see our own universe and the other universes in the pluriverse are going to remain hidden probably indefinitely. So it is hard to do science in relation to something that is never manifest to you. However there are some papers being written that attempt to say what can be said about the pluriverse based on what we know about our own universe.
Many things are hidden from us, like dark energy, dark matter, what was before the big bang, what is inside black holes, etc. So the fact that it is hidden is not a problem if we can deduce some trace that would indicate what the invisible thing is like, and David Deutsch’s experiment might well do the trick.
See also http://en.wikipedia.org/wiki/Many-worlds_interpretation