Global duality (as for instance on page 29 of Rubin's "Euler systems") gives you a description of the cokernel of your inclusion. Let $p$ be a prime. Then the $p$-primary part of the quotient of $Ш(E,S)$ by $Ш(E,S)$ is dual to the cokernel of the map
$$ \mathfrak S_p(E/k) = \varprojlim_n \,\mathrm{Sel}_{p^n}(E/k) \ \to \ \bigoplus_{v\in S} E(k_v)^{*}.$$
Here the source is the compact $p$-adic Selmer group of $E/k$, which is a finitely generated $\mathbb{Z}_p$-module of rank $r$, believed to be the rank of $E(k)$. The target is the sum of the $p$-adic completions of the local points $E(k_v)$. If $v$ is not above $p$, then this group is finite and otherwise it is a finitely generated $\mathbb{Z}_p$-module of rank $[k_v:\mathbb{Q}_p]$. 

One can make quite precise conjectures as to what the corank of the $p$-rprimary part of $Ш(E,S)$ should be in terms of the rank of $E(k)$ and $k/\mathbb{Q}$ by the above duality. Roughly speaking some sort of independence of elliptic logarithms tells you that the above map should have image as large as possible. For instance over $k=\mathbb{Q}$, it is clear that the corank of the $p$-primary part of $Ш(E,S)$ will be $0$ is $p$ if is not in $S$ or if $r\geq 1$, and $1$ otherwise. For larger $k$ it is a bit harder to formulate as it will depend on the field of definition of $E$, but generally speaking it tends to be large as soon as a $p$-adic place of large degree is in $S$.