Let me work in the category of $L^2$-automorphic representations. Assuming your global representation $\pi$ is irreducible, about the only thing you can say about an arbitrary local component $\pi_v$ is that it is an irreducible smooth representation of $G_v$. For almost all $v$, you can also say $\pi_v$ will be spherical.

I don't know exactly which groups you include in classical groups (unitary groups? non-quasi-split forms?) but if you allow compact groups then the trivial representation is cuspidal, and in positive rank it is locally non-generic everywhere.

For (split) SO(5) you have Saito-Kurokawa lifts, which are locally non-tempered. Also, many cuspidal representation of SO(5) are not generic everywhere. See references on Siegel modular forms, SO(5) or GSp(4).

As mentioned in comments, $\pi_v$ can typically only be supercuspidal or discrete series at a finite number of places (e.g., if G is GL($n$) with $n > 1$).