I'm going to post from the perspective of a natural scientist, so please forgive any misuses of philosophical terminology.
In thermodynamics, we define a spontaneous process as one that occurs without an external input to a system—that input is associated with energy, but isn't exactly the same (the technical term is free energy, but exactly what that looks like depends on the environment around your system in question).
This is important because it suggests a subtle frame challenge to the premise of your question: Spontaneous events have a cause, it's just internal to the system rather than external. Your example of radioactive decay is certainly spontaneous, but the cause is easy to identify: the configuration of such and such a nucleus has more energy than a different configuration, which can be accessed by emitting some particle (the decay event). Other answerers have also clarified that such an event is probabilistic: we cannot say with certainty when it will occur, merely that it can do so (and spontaneously: by itself).
All kinds of chemical reactions occur spontaneously, assuming the conditions are set up correctly beforehand. For instance, the conversion of organic polymers such as hydrocarbons and proteins into water and carbon monoxide is spontaneous in a sufficiently concentrated bath of oxygen gas and above some temperature known as the autoignition point. (This reaction is known as fire.) Iron spontaneously—but rather slowly—reacts with oxygen in the air, even at ambient temperature, to form iron oxide (that is rust).
Now, you might argue that setting up the conditions for a spontaneous process counts as an external input! You can probably expand the definition of "process" to find where the whole setup is spontaneous again, but the problem persists. Ultimately, of course, we reach a question beyond physics: what caused the universe to come into existence? After that, the Second Law of Thermodynamics suggests that every* process in the universe is spontaneous, slowly using up the available free energy. But this is only an (ironclad) empirical law, not a mathematical theorem.
One might say that a freely chosen action by a person is spontaneous because was internal to that person. But this is only an analogy, of course. Ultimately, even causality cannot be explained by the laws of physics.
* Or, at least, so close to every that it doesn't make a difference