Let $X$ and $Y$ be closed complex subvarieties in an ambient
complex manifold $M$. We will explain the intersection formula
$$c(X) \cdot c(Y)= c(TM)\cap c(X\cap Y)$$
for suitable notions of Chern classes and transversality for singular spaces.
If $X$ and $Y$ intersect transversal in a Whitney stratified sense, this is
true for the MacPherson Chern classes (of adopted constructible functions).
If $X$ and $Y$ are "splayed" in the sense of Aluffi-Faber, then this formula
holds for the Fulton-(Johnson-) Chern classes, and is conjectured for the
MacPherson Chern classes.
We explain, that the version for the MacPherson Chern classes is true under a micro-local "non-characteristic" condition for the diagonal embedding of $M$ with respect to $X\times Y$. This notion of non-characteristic is weaker than the Whitney stratified transversality as well as the splayedness assumption.