We first show that for an infinite dimensional Banach space
$X$, the unitary spectrum of any superstable operator is
countable. In connection with descriptive set theory, we show
that if $X$ is separable, then the set of stable operators and
the set of power bounded operators are Borel subsets of $L(X)$
(equipped with the strong operator topology), while the set
$\mathcal{S}'(X)$ of superstable operators is
coanalytic. However, $\mathcal{S}'(X)$ is a Borel set if $X$
is a superreflexive and hereditarily indecomposable space. On
the other hand, if $X$ is superreflexive and $X$ has a
complemented subspace with unconditional basis or, more
generally, if $X$ has a polynomially bounded and not
superstable operator, then the set $\mathcal{S}'(X)$ is non
Borel.