The question is probably two-folded and I will try not to make it too vague, but nonetheless the question remains general.
First fold:
In most physical laws, that we have analytic mathematical expressions for, one comes across functions that diverge at a given point, typical examples would be the Coulomb or the gravitational forces being $propto 1/r^2,$ clearly they diverge at $r=0.$
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Physically it is obvious that if by distance $r$ we mean the distance between center of masses of the objects, then $r=0$ is trivially excluded (for macroscopic objects at least) because they have well defined excluded volumes and cannot occupy the same space at the same time, hence one may argue that the divergence at $r=0$ case is a mathematical artifact and is to be ignored, but is this really the case or do we have an explanation for such extreme cases?
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Are most singularities met in classical physics just reminders of the fact that within classical models, not all can be explained, and one has to turn to more general frameworks such as QM, where then the singularities would be resolved?
Second fold:
The second type of singularity that one comes across, is in statistical mechanics or thermodynamics, namely the association of phase transitions to singularities of the free energy of the system. We know that if the nth order derivative of the free energy becomes singular then the system must at some critical point exhibit an nth order phase transition, or conversely if the free energy never becomes singular, e.g. if $F(T) propto frac{1}{T},$ then there can be no phase transition that depends on temperature as such function would only diverge at $T=0 K$ which physically is impossible anyway.
Typical examples would be second order phase transition in the Ising ferromagnet system, where the second derivative of the free energy with respect to $T$ diverges at the critical temperature $T_c,$ at which point the system transitions from a paramagnet to a ferromagnet or the other way around. An example for first order transition would be liquid water into ice, where the transition is first order because the first order derivative of the free energy becomes singular. Furthermore there are also cases that the free energy derivatives diverge on change of density of the system instead of temperature.
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What is the main difference between such type of singularities met in phase transitions, compared to the previous ones mentioned in the first part?
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Finally why should a phase transition correspond to a singularity in the free energy or entropy at all? What is the physical intuition here?
Feel free to use any mathematical argumentation you find necessary, or other examples that may find more illustrative.