1) The classification of p-groups (even p-groups of class 2) **is indeed wild** over $\mathbb{F}_p$ [V. Sergeichuk, The classification of metabelian p-groups (Russian), Matrix problems, Akad.
Nauk Ukrain. SSR Inst. Mat., Kiev, 1977, pp. 150-161.]

(It's not clear to me why Hamkins's answer on your other question led you away from asking about wildness...)

2) Rather than smoothness - since as pointed out by others, classification problems on finite spaces are trivially smooth - in addition to wildness, you might consider **computational complexity** as a measure of difficulty of a finite classification problem. In particular, the existence of a polynomial-time algorithm for a classification problem is some evidence that the classification is "easy."

There is currently no known poly-time algorithm to test isomorphism of finite groups, given by their multiplication tables, and it is widely believed that p-groups (even those of class 2) are the hardest cases. (Beware of potential circularity in beliefs though: the wildness of classifying p-groups of class 2 is one of the pieces of evidence for the preceding belief, although there are others as well.)

Furthermore, despite all of the exciting recent progress on classifying groups by coclass, as far as I know this hasn't yet been used to make any progress towards a polynomial-time algorithm for testing isomorphism of p-groups. Furthermore, isomorphism of p-groups of class 2 are believed to be the hardest cases, and these are exactly the groups of *maximal* coclass, for which the recent classification results say the least.

(Polynomial-time as a version of "easiness" is a little subtle compared to mathematical experience: for example, the fact that finite simple groups are all generated by 2 elements (a consequence of CFSG) implies that there is a relatively simple polynomial-time algorithm to test isomorphism of finite *simple* groups (find a generating pair of elements in the one group, try all possible maps of that pair into the other group and check if each is an iso), but that algorithm relies on a massive, difficult theorem for its correctness. However, I still think the computational complexity of a classification problem is one useful gauge of its difficulty, particularly when it comes to finite problems.)