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) homomorphisms 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 (EDIT … (thanks @Amitay for comments) for hyperspecial maximal compacts. For $\operatorname{GL}_n$, all maximal compacts are conjugate, and are hyperspecial. For $\operatorname{SL}_n$, they are all hyperspecial, but there are $n$ conjugacy classes. Every reductive group has at least one (conjugacy class of) hyperspecial maximal compact, but/and some do have non-hyperspecials: for example, the affine apartments of $\operatorname{Sp}_4$ have two different types of vertices, one with fewer edges touching it. The latter is not hyperspecial.
Principal series are generically irreducible. For classical groups, we can explicitly compute the $L^2$ norm (mod center) of the (essentially unique) spherical vector, (EDIT for hyperspecial maximal compact) and it's not finite….
Yes, there are some square-integrable subrepns of principal series, at some special values of the parameters. The archimedean case has the well-known examples of holomorphic discrete series subreps of principal series far away from the "unitary range".
