$p$-stability singles out $p = 0.5, 1, 2$. Specifically, there is no probability distribution $P$ such that the linear combination $\sum^n a_i X_i$ is distributed as $\|a\|_p Y$, where $X_1 ... X_n$ and $Y$ are random variables distributed according to $P$, if $p$ is not $1/2, 1, 2$.

$p$-stability singles out $0 < p \le 2$. Specifically, there is no probability distribution $P$ such that the linear combination $\sum^n a_i X_i$ is distributed as $\|a\|_p Y$, where $X_1 ... X_n$ and $Y$ are random variables distributed according to $P$, if $p$ is not in the range $(0, 2]$.

For $p = 0.5, 1, 2$ these distributions have closed-form expressions.

(note: updated to reflect Gideon Schectman's comment)