Many Worlds isn't taken seriously because there's "proof" of other universes - it's taken seriously because it's actually the simplest explanation mathematically. The equations of quantum mechanics naturally lead to superpositions (particles existing in multiple states). MWI just says "what if we don't add extra rules to make those superpositions collapse?" It's like if you have a math equation that gives you 5 answers, and instead of creating a complicated rule to pick just one answer, you just accept all 5. Thats why physisists consider it - parsimony.

None.

It is, however, actually simpler than other theories, in that if you just let quantum mechanics do it's thing without extra (unknown) parts to limit it, it produces many worlds. So, by Occam's razor...

None. There is absolutely no proof of many worlds or the multiverse. RE the god of the gaps. It's much more interesting to do physics rather than speculate about what falls outside the purview of the scientific method.

The “many worlds interpretation of quantum mechanics” is loosely that when you do a quantum coin flip, the universe splits into two universes, one for each result.

The reason for this thought is when you work with quantum mechanics, your system has a state that evolves smoothly, but if you “measure” it, the state suddenly snaps to (a random) one of the possible measurement values (when the coin isn’t being observed, it smoothly evolves, but once you measure it, it suddenly takes on a random value). However, if you expand your quantum description of the system to include your measurement device as well as the quantum “coin”, that sudden “snapping” goes away. Instead your whole system smoothly evolves, and it evolves into a “superposition” of the shared state of the state of the overall system in each of the possible measurement outcomes.

Extending this idea, it would seem that whenever you could describe a situation that acts like a “quantum coin flip”, both results happen, and the universe “splits”.

I really want to emphasize that the practical meaning of these “other worlds” is just that things are a lot “fuzzier” when you zoom in than classical statistics would suggest. Not that there’s another universe where you stayed with your ex or took a different career path or whatever.

Also this is an “interpretation” of quantum mechanics for good reason. It doesn’t really have any physical implications. In particular, it’s not possible to go “interact with” those “other universes”.

Most importantly, there are other “interpretations” of quantum mechanics, like that quantum mechanics is really a rethinking of statistics not of physics.

https://arxiv.org/abs/1301.1069

Effectively there is no current consensus on the issue, though Many Worlds and String Theory are widely regarded outside their adherents as non-falsifiable and therefore not legitimate theories.

Essentially the proponents of the theory have created beautiful math that fit their view, but absolutely nothing in the real world that can show that it is a more valid theory than any of the other theories which have equally elegant math to back them.

There isn't any "proof"; in fact, Many Worlds is what's called "unfalsifiable", which means we don't have a way through the scientific method to show Many Worlds to be false.

Also, it's not really

My God! There must be many worlds than just our one?

But more

There are moments in time where one path is taken and not another... but what if all paths are taken, somewhere?

It's not meant to be a valid theory, it's just a possible outcome of having a spacetime continuum; because it's not falsifiable though, it's not worth pursuing right now, only worth keeping in mind in case we come across new evidence to evaluate.