Examples of Stiefel-Whitney classes of manifolds

It goes back to Wu in the 1950's that if one can compute the mod 2 cohomology of a manifold, with its Steenrod operations, then one can explicitly compute its Stiefel-Whitney classes, via the Wu formula. See for example the Theorem on page 188 of ``A concise course on algebraic topology'' (no originality claimed, just the quickest reference for me to find).

Borel–Hirzebruch compute also $w(\mathbf G_2/\mathbf{SO}(4))$ and $w(G/S)$ ($G$ compact connected, $S$ toral subgroup) in Characteristic classes and homogeneous spaces (I, §17 and III, §5).

The total Stiefel-Whitney class of any connected compact orientable surface $M$ can be computed fairly easily. As $M$ can be embedded in $\mathbb{R}^3$ with trivial normal bundle, $TM$ is stably trivial and therefore $w(M) = 1$. Alternatively, $w_1(M) = 0$ as $M$ is orientable and $w_2(M) = 0$ as it is the mod $2$ reduction of the Euler class which is $(2 - 2g)a$ where $a$ is the generator of $H^2(M, \mathbb{Z})$.