Consider an $C^\infty$ abstract stratification $A$ (in the Thom-Mather sense, see Mather's note).
Can we embed $A$ in some $\mathbb{R}^n$ (or in an analytic manifold) as a subanalytic set?
If not, what could be the problem?
This apparently was shown by Laurent Noirel in his thesis for the $C^k$ case, $1 \leq k < \infty$.