Let $S$ be a $K3$ surface. Is it true that any sheaf on $S$ with zero Chern classes is isomorphic to $\mathcal{O}_S^{\oplus n}$ for some $n$? If not, do you have any counterexample?

The answer is no. Here is a counterexample. Take an ample divisor $L$ on $S$ and let $Z \subset S$ be a zerodimensional subscheme of length $\ell(Z)=L^2$. Now consider the coherent sheaf $$\mathscr{F}=\mathscr{O}_S(L) \oplus \mathscr{O}_S(L) \otimes \mathscr{I}_Z.$$ Straightforward computations show that $$c_t(\mathscr{O}_S(L))=1Lt, \quad c_t(\mathscr{O}_S(L)\otimes \mathscr{I}_Z)=1+Lt + \ell(Z) t^2,$$ hence $c_t(\mathscr{F})=1$. So $\mathscr{F}$ has zero Chern classes, but it is not isomorphic $\mathscr{O}_S^{\oplus 2}$ because it is not locally free. Note that this construction holds for any smooth projective surface $S$, in fact the assumption $K3$ is not used here. 

