Different QM interpretations say different things about wave function collapse. Roughly speaking, at a high level, there are three different general things that different interpretations say:

Some interpretations say that "wave function collapse" is something we do in the math to make correct predictions once we know the results of experiments, and doesn't reflect anything physical actually happening to an individual quantum system when it's measured. "Copenhagen" interpretations more or less take this approach (it's hard to be definite because "Copenhagen" is a very vague and general label for interpretations); so do statistical/ensemble interpretations such as the one used by Ballentine in his textbook. Intepretations in this category say that measurements do have single outcomes (unlike those in the third category below), but they offer no explanation of how that happens.

Some interpretations say that "wave function collapse" is an actual physical thing that happens to an individual quantum system when it's measured--in other words, during a measurement the actual dynamics of the wave function are not the same as the unitary dynamics it has between measurements. This is how such interpretations account for measurements having single outcomes. The main issue with this category of interpretations is that all attempts to construct an underlying model of what the dynamics would be during a measurement have failed.

Finally, interpretations like the Many Worlds Interpretation say that "wave function collapse" never actually happens, because measurements don't have single outcomes--all possible outcomes happen, each one in its own branch of the wave function. This is indeed what you get if you just apply unitary dynamics to the wave function all the time, including during a measurement. In interpretations like this, we ourselves, observing the results of measurements, have "branches" of our own wave functions, and when we think we've observed a measurement to have a single outcome, that's because "we" are just one branch of the wave function. In that particular branch, you can mathematically apply the "wave function collapse" rule to make predictions about what you, in that particular branch, will observe in the future, because the other branches will never interfere with yours.

So there is no single answer to your question; all we have are different QM interpretations that say different, mutually incompatible things.