The problem comes from Volumn I, (7.4.3), Grothendieck's EGA. For a scheme $X$, denote the sheaf of rational functions by $\mathscr{R}(X)$.
Corollary (7.4.3). Over an integral scheme $X$, any torsion-free, quasi-coherent and rank-$1$ $O_X$-module sheaf (denote this sheaf by $\mathscr{F}$) is isomorphic to a subsheaf of $\mathscr{R}(X)$. The converse also holds.
First, I think the statement should restric to nonzero sheaves. I know how to prove the direction "$\Rightarrow$".
On the other hand, assume $\mathscr{F}$ is a nonzero subsheaf of $\mathscr{R}(X)$, then I can show that it is torsion-free. If $\mathscr{F}$ is quasi-coherent, then I can also show that $\mathscr{F}$ is of rank-$1$. The key problem: how to prove $\mathscr{F}$ is quasi-coherent?
I know that subsheaves of constant sheaves may be non-constant. I think the corollary should be modified to the following.
New Statement. Over an integral scheme $X$, any torsion-free, quasi-coherent and rank-$1$ $O_X$-module sheaf (denote this sheaf by $\mathscr{F}$) is isomorphic to a nonzero and quasi-coherent subsheaf of $\mathscr{R}(X)$. The converse also holds.