  The structure of non-compactness of optimal Sobolev embeddings of $m$-th
order into the class of Lebesgue spaces and into that of all
rearrangement-invariant function spaces is quantitatively studied. Sharp
two-sided estimates of Bernstein numbers of such embeddings are obtained. It is
shown that, whereas the optimal Sobolev embedding within the class of Lebesgue
spaces is finitely strictly singular, the optimal Sobolev embedding in the
class of all rearrangement-invariant function spaces is not even strictly
singular.
