In the literature, a "smooth triangulation" seems to mean: a homeomorphism from a simplicial complex, such that on each simplex the map can be extended to a smooth map from a neighborhood of the standard simplex. Smooth manifolds always have them.

One could ask for more: that the map on each simplex can be extended to a diffeomorphism from a neighborhood of the standard simplex.

Are there non-cuspy triangulations of smooth manifolds?