  This paper investigates instances of Sobolev embeddings characterized by
local compactness at every point within their domain, except for a single
point. We obtain the sharp conditions that distinguish compactness from
non-compactness and observe that in the context of Sobolev embeddings,
non-compactness occurring at only one point within the domain could give rise
to an infinite-dimensional subspace where the embedding is invertible (i.e.,
not strictly singular). Furthermore, we establish lower bounds for the
Bernstein numbers, entropy numbers, and the measure of non-compactness.
