  Let $G$ be an infinite discrete group. A classifying space for proper actions
of $G$ is a proper $G$-CW-complex $X$ such that the fixed point sets $X^H$ are
contractible for all finite subgroups $H$ of $G$. In this paper we consider the
stable analogue of the classifying space for proper actions in the category of
proper $G$-spectra and study its finiteness properties. We investigate when $G$
admits a stable classifying space for proper actions that is finite or of
finite type and relate these conditions to the compactness of the sphere
spectrum in the homotopy category of proper $G$-spectra and to classical
finiteness properties of the Weyl groups of finite subgroups of $G$. Finally,
if the group $G$ is virtually torsion-free we also show that the smallest
possible dimension of a stable classifying space for proper actions coincides
with the virtual cohomological dimension of $G$, thus providing the first
geometric interpretation of the virtual cohomological dimension of a group.
