(1) Suppose that $X$ is a smooth complex algebraic variety, stratified by some nice smooth stratification $S$. Let $M$ be a $D$-module on $X$, s.t. its shriek-pullback (or star... whatever is comfortable to you) to every stratum is smooth. What can we say then about the singular support of this $D$-module? For example, I hope that every point in the singular support will be zero when evaluated on tangents to stratums. Maybe one will need $M$ to be holonomic R.S. for that.
(2) I want this in order to understand how to restrict this $D$-module to a subvariety, transversal to this stratification (say that upper-star and upper-shriek differ by a shift, and are in one cohomological degree).
So if anyone can explain this, or give some reference, I will be thankful.
Edit: I will be happy with an answer for (2) in the constructible world as well.