  In this article we introduce the space of configurations of commuting
elements in a topological group and show that it satisfies rational homological
stability for the sequences of unitary, special unitary and symplectic groups.
We also prove that it satisfies cohomological rational representation stability
with respect to the number of elements in the tuple for finite products of such
groups, in particular cohomological rational stability for the space of
unordered configurations of commuting elements. Finally we present some
computations of cohomology in the unstable range.
