Submodules of $\operatorname{Hom}_R(M,N)$ with $R$ a commutative ring. 
Is there a way to characterize the submodules of the $\operatorname{Hom}_R(M,N)$? 

$M,N$ are arbitrary $R$-modules and $R$ a commutative ring, to assure that $\operatorname{Hom}$ will be an $R$-module.
Someone optimistic would say that these submodules are in a bijective correspondence with the submodules of $N$, but I seriously doubt it. 
Edit. My question goes to if there is a relation between the submodules of $N$ and the submodules of $\operatorname{Hom}$, supposing you know the submodules of $N$ and $M$. 
 A: As I was commenting, the question in your comment and in your edit is more precise and can be answered, while the original question is quite more general and I cannot say if something can be said.
In general, the submodules of $M$ and $N$ are not enough to classify all the submodules of $Hom(M,N)$. 
For an easy example take a field $k$, $N=k$ and $M=\bigoplus_\mathbb N k$ (a vector space of countable infinite dimension). Then, $Hom(M,N)≅\prod_\mathbb N k$ has uncountably many submodules, so this information cannot be encoded in terms of the countable sets of submodules of $M$ and $N$. 
Let us remain for a while in a similar setting to the above example, that is, $k$ is a field, $M$ is any vector space and $N=k$. Then, it is possible to endow the $k$-vector space $M^*=Hom(M,k)$ with a linear topology which is called the pointwise convergence topology, which makes $M^*$ into a linearly compact topological vector space. It is a classical result that there is a bijective correspondence between the sub-vector spaces of $M$ and the closed sub-vector spaces of $M^*$ (this is a consequence of the so-called Lefschetz duality, which is a particular case of the MacDonald duality which works over complete local noetherian commutative rings). This can be seen as a result that goes in the direction you are interested in...
