Alexander's horned sphere is a topological sphere in 3-space that cannot be "ironed out", otherwise we would get a smooth (or PL) 2-sphere having a complementary region which is not simply-connected, a fact which is excluded because every smooth (or PL) 2-sphere in 3-space is standard.
Another simple topological object that cannot be smoothly ironed is a 2-dimensional disc inside $D^4$, obtained by coning over a knot in $S^3$.
Maybe such very pathological embeddings can be excluded a priori from the topological Whitney-embedding theorem, so it might be that this is these are not really an issue here.
