TITLE: Restrictions of Diffeomorphisms
QUESTION [7 upvotes]: Notation: Let$M$ be a smooth, closed manifold, $S$ any submanifold of $M$, $Diff(M)$ the group of diffeomorphisms of $M$ and $Imb(S, M)$ the group of smooth imbeddings of $S$ into $M$.
A classical result of R. Palais from the 1960 paper Local triviality of the restriction map for embeddings says that the map $Diff(M)$  $\rightarrow$ $Imb(S, M)$ given by restriction is a fibration.
I feel like I've heard during numerous teas that there are various refinements and generalizations of this due to J. Cerf and (possibly) others.
(1) Can anyone summarize what else is known in this direction beyond the theorem of Palais? 
(2) Is there a way to see Palais' result easily? [added: from the responses it sounds like the original paper is still a great way to see this result. But see the answers of Randal-Williams and Palais for an alternate route.]

REPLY [3 votes]: This is closely related to (and can be proved by the same methods as) the fact that if we fix another manifold $N$ then the restriction map
$$\mathrm{Emb}(M, N) \to \mathrm{Emb}(S, N)$$
is a locally trivial fibration, which was proved in the same paper by Palais. However, I wanted to advertise a geodesic proof of this fact:
E. L. Lima, On the local triviality of the restriction map for embeddings,  Commentarii Mathematici Helvetici Volume 38, Number 1, pp 163-164.