I am interested in the orders of random permutations. Since the law of the log of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), one expects that the probability for two permutations of $S_n$ to have the same order goes to 0 as n goes to infinity. Indeed experimentally this seems to happen with speed $O(1/n^2)$
I know that Wilf proved an asymptotic for a permutation in $S_n$ to be of order $d$ (https://www.math.upenn.edu/~wilf/website/Asymptotics%20of%20exp%28P%28z%29%29.pdf) but I don't think it can be used directly.
On the other hand it is clear that the probability that two permutations have same order is more than probability that two permutations are conjugate. This is $K/n^2$ according to Flajolet et al. (http://arxiv.org/abs/math/0606370), but here again I failed to generalize the method for the order.