Local linear Morsifications

The number of Morse points in a Morsification determines the topology of the Milnor fibre of a holomorphic function germ $f$ with isolated singularity. If $f$ has an arbitrary singular locus, then this nice relation to the Milnor fibre disappears. We show that despite this loss, the numbers of stratified Morse singularities of a general linear Morsification are effectively computable in terms of topological invariants of $f$.