The problem with this question is that it assumes there is some metaphysical interpretation that we can be sure is true. While we have excellent equations that work incredibly precisely, we are not sure which qualitative interpretation of these equations is real.
There are now countless interpretations, each with their own sub-interpretations. Alexander R. Pruss splits these interpretations into two main groups - No collapse theories with a deterministically evolving wavefunctions and wavefunction collapse theories.
Out of the collapse theories, we have the Copenhagen Interpretation, where the wavefunction collapse is triggered by a measurement. Definitions of what constitutes a measurement can differ a lot depending on the physicist/philosopher. The Ghirardi-Rimini-Weber theory is another collapse theory where the collapse is triggered at some particular rate over time. The trouble with this theory is that no spontaneous collapse has been observed in any way, and an additional parameter - that of the rate of collapse - has to be introduced and explained in some way.
There are also many no collapse theories such as Bohmian Mechanics, the Many Worlds Interpretation, Many Minds Interpretation and Traveling Forms interpretation. In these, the universe continues to develop deterministically, but each have their own reasons as to why we can only get stochastic results from the deterministic systems upon measurement. Each of these interpretations also have their own problems. Bohmian Mechanics has the problem of nonlocality. The Many Worlds Interpretation is unclear as to how splits occur and is a bit bizarre to try to reconcile with, for example, the conservation of energy. The Many Minds interpretation leads to bizarre absurdities such as Boltzmann Minds and universes where there is just one mind surrounded by zombies. I don't think the Traveling Forms is well enough known to have its own critique, but I expect someone will come up with one at some point.
I found an excellent study of this topic in this book: http://www.michalpaszkiewicz.co.uk/blog/reviewnapocs/index.html