What do characteristic classes of spinor bundle depend on?

Question

Let $M^n$ be a smooth manifold. Equip $M$ with a Riemannian metric and let $S$ be a spinor bundle. We can consider characteristic classes of $S$ (or $S_+,S_-$ for when $n$ is even), for example the Euler characteristic. My question is: Will these classes depend on the metric $g$ or spin structure chosen and if not, are they expressable in terms of other topological invariants of $M$?