My experience seems to be that quite often "generic" mathematical objects tend to be either extremely well behaved or structured, or at the opposite extreme are as unstructured as possible.
For example, random finite graphs have good expansion properties but actually constructing explicit families of expanders is quite challenging. There are a number of theorems that state that generic finitely generated subgroups of certain classes of topological groups are free.
On the other extreme, nearly all finite groups are (believed to be) $2$-groups (even $2$-step solvable), which is in some sense as unstructured as a finite group can be (in particular, I think nobody feels one can classify them up to isomorphism). Generic finite semigroups are (believed to be) $3$-nilpotent, meaning they have a multiplicative zero and the product of any $3$ elements is $0$. This again would indicate that the generic object is highly non-structured.
I am interested in other examples in different areas of mathematics of this phenomenon of generic objects being unreasonably structured or simply unreasonable.
As usual for big list questions, please provide one example per answer.