Given a scheme $X$ and an $\mathcal{O}_{X}$-module $\mathscr{E}$, we know that a section $s \in H^{0}(X, \mathscr{E})$ is equivalent to a morphism $s :\mathcal{O}_{X} \to \mathscr{E}$. It is the cokernel of this map which we typically call the cokernel of the section $s$.
For a line bundle $L$ and a section $s \in H^{0}(X, L)$ we know that the vanishing locus of $s$, denoted $Z(s)$, is a divisor. I believe the cokernel of $s$ is isomorphic to $L|_{Z(s)}$. In other words, one's intuition should be that the cokernel of a section of a line bundle encodes the restriction of the bundle to its vanishing locus. If I'm not mistaken, the same holds for locally-free sheaves of higher rank (EDIT: is this actually true?).
But I'm quite confused about the cokernel of a section $s$ of a more general $\mathcal{O}_{X}$-module $\mathscr{E}$ (let's assume coherent or quasi-coherent if it helps). All I know for sure is that since the cokernel is a quotient of $\mathscr{E}$, its support is contained in the support of $\mathscr{E}$. One can still make sense of the vanishing locus $Z(s)$ (EDIT: as the locus where $s$ vanishes in the fiber $\mathscr{E}_{x} \otimes k(x)$) although I feel like this is not an algebraic set in general (of course the support of a coherent sheaf is a closed subscheme, not to be confused with vanishing locus of a section). So the intuition from line bundles that the cokernel encodes data on the vanishing locus can't generalize exactly.
Can someone maybe spell out what we can say in general here, or must it be done on a case-by-case basis? I'd appreciate any intuition on how one should think of these cokernels.
