  If a complex analytic function, $f$, has a stratified isolated critical
point, then it is known that the cohomology of the Milnor fibre of $f$ has a
direct sum decomposition in terms of the normal Morse data to the strata. We
use microlocal Morse theory to obtain the same result under the weakened
hypothesis that the vanishing cycles along $f$ have isolated support. We also
investigate an index-theoretic proof of this fact.
