It is a fascinating concept indeed!
One approach is to study the cosmic microwave background radiation, which is the residual heat left over from the Big Bang. By analyzing the patterns and fluctuations in this radiation, scientists hope to find clues about the early universe and potentially uncover evidence of other universes beyond our own.
Honestly, I would not believe any results coming from that in the next decade. There are some spurious signals in the cosmic microwave background, and one possible explanation would be the existence of multiverses (or, to be more specific, the existence of a quantum-mechainical probability wavefunction that spans all multiverses). However, there are dozens of alternative hypotheses that could explain these signals (that I am aware of, there are probably hundreds more), and it is extremely hard to rule them out, since all we can do is look at it, we can not perform any experience.
Many interpretations of the extremely weird results in quantum mechanics infer the existence of a multiverse, but they are interpretations, there is not really any proof. The same goes for string theory, which is even less reliable, since the theory itself has never been tested, whereas we know quantum mechanics to be true and accurate.
Personally, I believe in the existence of a multiverse, but I doubt that we will prove or disprove this in our lifetimes. It is a very cool thought, though, and it explains a lot of conundrums (for example, why are the natural constants so finely tuned as to allow for the existence of complex elements and, ultimately, life? The answer would be: We live in one of the few Universes where this is true, and there are many more which do not host life).
Before we can reliably say that we have detected signatures of multiverses we have to understand our own Universe, and we are still far away from that. We can not explain Dark Matter or Dark Energy, and our most successful theories (quantum mechanics and general relativity) don't like each other on small scales... Also, our standard model of particle physics is incomplete. Until we have figured out these things, we can never really say whether a signal is a signature of a different Universe, or just something that we don't understand about our own...