What really causes radioactivity? We know that a nucleus can transform itself by many ways such as fission, alpha decay, beta decay, etc.
But what is it that propels this to happen?
Is it just the instability of the nucleus in general? So in that case, the idea I have is that the redistribution of the available forms of energy(mass-energy, binding energy, also some kinetic energy of the by-product) would make it more stable?
Or is there a better idea/statement in general?
 A: Nuclei belong to the quantum physics domain, which obeys different laws and postulates than the classical domain. Nevertheless, there is mathematical continuity  when going from the quantum domain to the classical, and vice verso.
In the classical domain there is the second  law of thermodynamics that translates into the minimum energy principle.

The principle of minimum energy is essentially a restatement of the second law of thermodynamics. It states that for a closed system, with constant external parameters and entropy, the internal energy will decrease and approach a minimum value at equilibrium.

This, in the framework of quantum dynamics translates into the statement by G.Smith in the comments , :" In general, in quantum physics everything that can happen does happen, with some probability. "
In quantum mechanics there are a lot of conservation rules, and the "can happen" refers to that. Except energy, momentum and angular momentum conservation, there are quantum number conservations which were imposed so that the data could be explained. Quantum mechanics had to be invented because of the quantized nature of light, seen in the black body radiation and the photoelectric effect, and the periodic table of elements which has stable nuclei and a logic not accessible to classical equations.
So in quantum mechanical systems, if there exists a lower energetically allowed by conservation rules  state, the principle of minimum energy makes it inevitatble that the nucleus will eventually settle at that ground state. Eventually refers to the probability nature of quantum mechanics, and the probabilities can be calculated using the known forces affecting the elementary particles, strong, weak, electromagnetic. Strong means high probability of falling to lower energy states if allowed,  as in fission,fast. Weak takes the time of weak interactions as in the beta decays, electromagnetic is faster than weak and slower than strong,as in gamma decays.
A: Radioactivity is one of the textbook examples of the inherent randomness of nature that cannot be reduced or explained by classical arguments. It is a quantum mechanical phenomena. 
Classical mechanics is only an approximation. The fundamental description of reality is quantum mechanics.
In short, trying to model the randomness of the nuclear decay by some hidden predictable classical dynamics, is impossible.    
