Let $S$ be a $K3$ surface and $H$ be an ample line bundle on it. Given a flat family of coherent sheaves on $S$ whose generic point is a $\mu\_H$-stable vector bundle, what can I say about the non-generic points? Are they always $\mu\_H$-semistable and torsion free? I think that this is equivalent to requiring the existence of a compactification of the moduli space of $\mu\_H$-stable vector bundles with fixed Chern classes, whose boundary points are $\mu\_H$-semistable torsion free sheaves. 



Limit of stable vector bundles.
I think the answer to your question is **no**, because of the following example. For the details, see [Huybrechts-Lehn, The geometry of moduli space of sheaves, Section 5.3].


Let $\pi \colon X \to \mathbb{P}^1$ be an elliptic $K3$ surface with irreducible fibres and assume that there is a section $\sigma \subset X$. By adjunction $\sigma$ is a $(-2)$-curve. Let $f$ be the class of a fibre of $\pi$ and set $H= H\_m:= \sigma + (m+3) f$. 


Then the Nakai-Moishezon criterion implies that $H$ is ample for $ m \geq 0$. Moreover, for $m >>0$, the $\mu\_{H}$-semistability of a rank $2$ vector bundle $E$ such that $c\_1(E) \cdot f \equiv 1$ (mod $2$) is equivalent to its $\mu\_H$-stability.


Now let us consider an extension of the form


$$0 \to \mathcal{O}\_X (f) \to E \to \mathcal{O}\_X( \sigma - 2f) \to 0.$$


One proves that this extension is $\mu\_H$-stable if and only if it does not split. Moreover, $c\_1(E) \cdot f = (\sigma -f) \cdot f=1$, so $\mu\_H$-stability and $\mu\_H$-semistability are equivalent in this case (at least if $m$ is large enough).


Finally, $\textrm{Ext}^1(\mathcal{O}\_X( \sigma - 2f), \mathcal{O}\_X (f) ) \cong \mathbb{C}^2$, and the zero element of this cohomology group corresponds to the trivial extension. So if one takes a disk $\mathcal{D} \subset \mathbb{C}^2$ centered at the origin, there is a flat family 


$$\{E\_t\}\_{t \in \mathcal{D}}$$


of sheaves on $X$ such that $E\_t$ is $\mu\_H$-stable for $t \neq 0$, whereas $E\_0=\mathcal{O}\_X (f) \oplus \mathcal{O}\_X( \sigma - 2f)$ is $\mu\_H$-instable. In fact, the inclusion $ \mathcal{O}\_X( \sigma - 2f) \hookrightarrow E\_0$ provides a destabilizing sub-bundle of $E\_0$. 


