The controversy over the interpretation of quantum is, confusingly, not about interpreting the equations of quantum theory. An interpretation would describe how those equations describe reality. The Copenhagen and statistical interpretations say that those equations don't describe reality, but they provide no alternative and keep using the equations anyway. They would add a collapse postulate but give no explanation of how or collapse happens. As such, their predictions are vague and ad hoc and there's no reason to consider them further.
The pilot wave theory takes the equations of motion of quantum mechanics and supplements them with an additional equation describing the motion of particles in addition. Under some circumstances those particles are supposed to obey different probability rules than quantum theory:
https://arxiv.org/abs/1906.10761
Since this theory has different predictions than quantum theory it isn't an interpretation of quantum theory, it's a candidate to replace quantum theory.
There are theories called spontaneous collapse models that modify the equations of quantum theory to include collapse to produce one specific outcome:
https://arxiv.org/abs/2310.14969
These theories also have different predictions than quantum theory and so are an alternative to quantum theory, not an interpretation of it.
It should be noted that pilot wave and spontaneous collapse theories can't currently reproduce the predictions of quantum field theories that are used to make the vast bulk of actual predictions made using quantum theory:
https://arxiv.org/abs/2205.00568
The many worlds interpretation (MWI) takes the equations of quantum theory as a description of how reality works and works out their consequences. In this theory each measurable quantity has multiple possible values and those values can intefere with one another. When information is copied out of a system that suppresses interference between the different possible values of the measured quantity, this process is called decoherence:
https://arxiv.org/abs/quant-ph/0306072
As a result of this process, on the scale of everyday life reality as described by quantum theory looks a bit like a collection of parallel universes:
https://arxiv.org/abs/1111.2189
https://arxiv.org/abs/quant-ph/0104033
When you do a measurement, the result you see won't interfere with other versions of the same measurement as a result of decoherence so you will only see one of them. As a result you can adjust the state to reflect that lack of interference:
https://arxiv.org/abs/2008.02328
This is why the MWI was originally called the relative state interpretation, the state is relative to what outcome happens in your branch. The collapse rule works to the extent that it works at all just describes resetting the relative state to take account of measurement outcomes.
I should note that one of the answers above states that the MWI discards the Born rule. This is misleading. The standard quantum probability rule has been explained by a couple of different arguments in unmodified quantum theory:
https://arxiv.org/abs/0906.2718
https://arxiv.org/abs/quant-ph/0405161
So the MWI currently provides the only known explanations of how the probability rule arises and of the circumstances under which it works: it requires decoherence.