Let $X$ be a projective variety, and $E$ be a coherent sheaf on $X$. Grothendieck has proven that there is a scheme $\mathrm{Quot}_X(E)$ parametrizing arbitrary quotient sheaves of $E$. It is probably a well-known question, but I found nothing in Google:

$$\text{Why not a scheme $\mathrm{Sub}_X(E)$ parametrizing arbitrary subsheaves of $E$?}$$

Does such a scheme exist? If yes, is it purely a question of style, like Grothendieck's definition of the projectivization $\mathbb P(V)$ as the set of one-dimensional quotients of a vector space $V$? If not, is it because cokernels of sheaves require sheafification, while kernels coincide with kernels for pre-sheaves?

Moreover, if one considers a reflexive sheaf $E$ (that is the canonical morphism $E \to E^{**}$ is an isomorphism) and its reflexive subsheaves, then there is a Sub-scheme via dualizing and taking a Quot-scheme. Is there a less restrictive constraint?

locally freequotients. I guess that the answer is then that there are more locally free quotients than locally free subobjects. $\endgroup$ – HeinrichD Nov 7 '16 at 10:37proofof existence of Quot-schemes you'll see exactly how quotients are better-suited than subsheaves (related to Mohan's comment). $\endgroup$ – nfdc23 Nov 7 '16 at 16:22