Can one always extend a smooth triangulation from a smooth submanifold $S$ to the ambient manifold $M$? (For simplicity both $S$ and $M$ are compact without boundary). Is the extension possible when $S$ is a sphere?

What I know is summarized in this answer. Namely, in the PL category the answer is yes, by a result of Armstrong, i.e., one can always extend a triangulation from a PL submanifold. Also one can always extend a smooth triangulation from a codimension one separating hypersurface (this is due to Munkres). I think, the same is true for non-separating hypersurfaces. Thus it is enough to extend the smooth triangulation of $S$ to a tubular neighborhood of $S$.

Naively, I would expect that the extension is always possible. Otherwise, one can probably use it to define a potentially interesting invariant of smooth manifolds, and I have never heard of such an invariant.

Can one always extend a smooth triangulation from a smooth submanifold $S$ to the ambient manifold $M$? (For simplicity both $S$ and $M$ are compact without boundary). Is the extension possible when $S$ is a sphere?

What I know is summarized in this answer. Namely, in the PL category the answer is yes, by a result of Armstrong, i.e., one can always extend a triangulation from a locally flat PL submanifold. Also one can always extend a smooth triangulation from a codimension one separating hypersurface (this is due to Munkres). I think, the same is true for non-separating hypersurfaces. Thus it is enough to extend the smooth triangulation of $S$ to a tubular neighborhood of $S$.

Naively, I would expect that the extension is always possible. Otherwise, one can probably use it to define a potentially interesting invariant of smooth manifolds, and I have never heard of such an invariant.