As it happens, (admissible) supercuspidals cannot be spherical, because (by Borel-Casselman-Matsumoto) admissible repns with Iwahori-fixed vectors have non-trivial maps to and from unramified principal series. The Jacquet-module vanishing condition for supercuspidals is exactly that (via Frobenius Reciprocity, etc.) they admit no (non-zero) homs to principal series.

Many people would count supercuspidals as discrete series.

For p-adic groups, I myself do not know much about discrete series that are not supercuspidals. 

And, again, if a discrete series repn were spherical, it would be a sub and quotient of principal series, which is not possible.

Principal series are generically irreducible. For classical groups, we can explicitly compute the $L^2$ norm (mod center) of the (essentially unique) spherical vector, and it's not finite...