I would like to be able to compute the Stiefel--Whitney class of an associated bundle: if $V$ is a representation of $G$ and $P\to M$ is a principal $G$-bundle, then $$\frac{P\times V}{G}\to M$$ becomes a vector bundle over $M$.
I'm particularly interested in case where $P\to M$ is the principal $H$-bundle $G \to G/H$ (where $G$ is a compact Lie group and $H$ is a closed, connected subgroup), and $V$ is the adjoint representation of $H$ on $\mathfrak g/\mathfrak h$. In this case, the associated bundle is diffeomorphic to the tangent bundle $T(G/H)$. If I understand correctly, I know for other reasons that $G/H$ should be orientable, but I would like to prove it by computing the first Stiefel--Whitney class of $T(G/H)$.
Let $Ad: H \to GL(\mathfrak{g}/\mathfrak{h})$ be the adjoint representation. Basically this this could answer your question. To be more precise: The principal bundle $G \to G/H$ has a classifying map $$ f : G/H \to BH$$ Composing this map with $BAd:BH \to BGL(k)$ will give the associated $GL(k)$ principal bundle. For these objects Stiefel-Whitney classes are defined in $H^*(BGL(k);\mathbb{Z}/2)$ and we have $w_i(G \to G/H) = f^* BAd^* w_i.$ Explicelty the first Stiefel-Whitney class is induced by the determinant $$ B\det: BGL(k) \to B\mathbb{Z}/2.$$ Since $Bdet \circ BAd = B(det\circ Ad)$ and $det \circ Ad \equiv const$ this gives $$w_1(G\to G/H) = f^* \underbrace{BAd^*Bdet^*}_{=0}x = 0$$ where $x \in H^1(B\mathbb{Z}/2,\mathbb{Z}/2) $ is the generator. But the whole point is that the adjoint representation takes values in orientation preserving automorphisms.