This is a continuation of A question about regularity of foliations>this question answered by Dmitri.

Let $F$ and $F'$ be smooth ($C^\infty$) foliations of a manifold $M$. Assume that there is a homeomorphism $h$ that takes $F$ to $F'$.

- Assume $F$ has a compact leaf, can one conclude there is a diffeomorphism that takes $F$ to $F'$?
- Assume that $h$ is $C^1$, can one conclude there is a ($C^\infty$) diffeomorphism that takes $F$ to $F'$?
- Assume that $M$ is a torus and $F'=L$ is a Diophantine-irrational-line foliation of the torus. Then, in case of $M=T^2$ following the outline of Sam Nead one can show that there's a diffeomorphism that takes $F$ to $L$, does this hold for higher dimensional tori?
- Dmitri gives us certain obstruction to existence of a diffeomorphism. Are there other interesting obstructions?