  We prove an analogue of Miller's stable splitting of the unitary group $U(m)$
for spaces of commuting elements in $U(m)$. After inverting $m!$, the space
$\text{Hom}(\mathbb{Z}^n,U(m))$ splits stably as a wedge of Thom-like spaces of
bundles of commuting varieties over certain partial flag manifolds. Using
Steenrod operations we prove that our splitting does not hold integrally.
Analogous decompositions for symplectic and orthogonal groups as well as
homological results for the one-point compactification of the commuting variety
in a Lie algebra are also provided.
