The compactness criterion of frechet kolmogorof gives necessary and sufficient conditions on when a set in $L^p$ is compact.
Given a set of function in a Sobolev space $W^{k,p}$ one can apply that theorem to the set and all its weak derivatives up to order $k$ to obtain compactness.
I wonder if there is a established theorem that gives compactness criterions on a subset of a fractional sobolev space.