It's difficult to know how an argument against the principle could begin. As humans, we seem to have a deeply ingrained model of the universe that implies all events have causes. Our instinct seems to be to assume a cause without having any explicit evidence that a cause can even exist. Consider the case of the the beginning of the universe. If any event is likely to be causeless, it is that event. And yet, there exist any number of theories that attempt to explain the Big Bang. Our intuition that everything has a cause seems to literally have no bounds—not even the universe can contain it.
One avenue of attack would be to suggest that our model of causation was itself uncaused and therefore it is not reliable. But that naturally leads us to question how we can make an argument against the idea of causation based on the principle of sufficient causes. (I am reminded of Plantinga's concept of defeaters here.) And more damaging, the argument, if it succeeds, merely shows that we can't trust our intuition, not that our intuition is false. It would be an attack on the epistemological question, not the metaphysical one.
We aren't asking about Determinism, which says that if we know the current state of the universe and the rules that govern it we can (in theory) know every other state of the universe. There are certainly good arguments against that hypothesis. And if we could find an argument against the "principle of sufficient reason", we could debunk determinism easily enough. (If things just happen, we can't very well predict them.)
But showing that determinism is a bad model for reality has no bearing on this question at all. If I find a coin on the ground with heads showing, there are any number of ways it could have gotten there. But since we all accept the principle of sufficient reason, we all agree that something must have caused the coin to be there and we all reject the idea that coins spontaneously appear on the ground. Nor is it a problem that the coin is showing heads rather than tails because there exist approximately equal number of causes that result in that state as opposed to the other. A coin carefully balanced on its edge excludes a number of causes, but we are certain that we will eventually find some set of causes that result in that state even if can never be sure which particular cause actualized it.
Quantum mechanicsQuantum mechanics is a model of parts of the universe that suggests a number of counter-intuitive results, but as far as I can tell people who explore the model still expect to discover some set of causes for everything they observe. A simple test of that assertion is to imagine what will happen if a scientist notices something that the theory does not predict. They will likely redo the experiment, reinterpret their results, adjust the theory, or some combination of the above. What they won't do is say, "Oh well. Things sometimes happen that don't have any reason at all to happen."
There's no evidence that disproves the principle of sufficient reason (and precious little that proves it), so we can continue to behave as if it is true without fear of behaving irrationally.