Stiefel classes and generic sections One can say that the Stiefel-Whitney classes is dual classes to the locus of linearly dependence of generic sections. What means "generic"? It would be great, if you show me some relation in local coordinates.
The same question about the Chern classes and a complex generic sections.
 A: I have searched the internet a little bit and I can give a partial answer, phrased in the terms of the Euler class. I hope it will be useful to you. 
In these notes of Michael Hutchings it is proved (Theorem 5.2) that if $E \rightarrow M$ is a smooth, oriented, rank $n$ vector bundle over a smooth, closed, oriented manifold $M$, then the Euler class $e(E) \in H^{n}(M, \mathbb{Z})$ is Poincare dual to the zero set $s^{-1}(0)$ of a generic section $s: M \rightarrow E$. Here, generic means the section $s$ is transversal to the zero section. You can throw away all orientability assumptions and work exclusively with $\mathbb{Z} _{2}$ coefficients, in fact, the proofs given in the notes seem to work and in fact become simpler, since there is no problem with the signs. 
Now, if $E \rightarrow M$ is a real line bundle, then this already solves our problem, since in this case we have $e(E) = w _{1}(E) \in H^{1}(M, \mathbb{Z}_{2})$. We can extend this result if we work a little harder, by relating Euler classes and Stiefel-Whitney classes.
If $E \rightarrow M$ is a real rank $n$ vector bundle, then we have $w_{1}(E) = w_{1}(det(E))$, where $det(E) = \Lambda ^{n} E$ is the determinant bundle of E. Hence, also in this case we can define $w_{1}(E)$ as being Poincare dual to zeroes of some generic section, although in this case it is not a section of $E$ (but of $det(E)$).
(More generally, in prof. Kreck's Differential Algebraic Topology, chapter 17, it is shown how one can define Stiefel-Whitney classes using only the Euler class, by looking at the coefficients of $e(p _{M}(E) \otimes p_{\mathbb{RP}^{N}}(L))$ in the Kunneth decomposition of $H^{n}(M \times \mathbb{RP}^{N})$, where $p_{M}, p_{\mathbb{RP}^{N}}$ are the projections from the product $M \times \mathbb{RP}^{N}$ to its factors, $L \rightarrow \mathbb{RP}^{N}$ is the tautological bundle nad $N$ is "large enough".)
