Supercuspidal, spherical and discrete series representation Let $G$ be an algebraic group defined over $\mathbb{Q}$ with maximal unipotent radical $N$. Let $\pi$ be an admissible representation of $G(\mathbb{Q}_p)$, we say that this representation is supercuspidal if $\pi/\langle\pi(n)v-v\rangle = 0$. This condition is equivalent to the fact that their matrix coefficients have compact support modulo $Z(\mathbb{Q}_p)$, the centre of $G(\mathbb{Q}_p)$.
Let $K$ be a maximal compact subgroup of $G(\mathbb{Q}_p)$, we say that $\pi$ is spherical or unramified if $\pi^K$, the space of $K$-fixed vectors of $\pi$ has dimension bigger than $0$.
We say that $\pi$ is a discrete series representation if the matrix coefficients of $\pi$ are $2$-integrable modulo $Z(\mathbb{Q}_p)$.
Usually I have found that people divide the admissible representations into three disjoint sets: Supercuspidals, non supercuspidals and discrete series and spherical. Is this decomposition true? Can a supercuspidal representation be spherical? (If not, why not?) Can a discrete series be spherical? (If not, why not?) Why the classical principal series representation (induction of characters of the Torus) are not discrete series?
 A: 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. At least every (EDIT: thanks @LSpice) reductive group that splits over an unramified extension has at least one (conjugacy class of) hyperspecial maximal compact, but/and some (EDIT: split groups!) 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".
