TITLE: Examples of Stiefel-Whitney classes of manifolds
QUESTION [9 upvotes]: Let $M$ by an compact, connected $n$-dimensional manifold without boundary. 
Are there any other computable examples of the Stiefel-Whitney class $w(M)$ except for $M=S^m, \mathbb{R}P^m,\mathbb{C}P^m, \mathbb{H}P^m$?

REPLY [3 votes]: The manifold $P(m,n) := (S^m\times \mathbb{CP}^n)/(\mathbb Z/2)$, where $\mathbb Z/2$ acts by the antipodal map on $S^m$ and by complex conjugation on $\mathbb{CP}^n$, is called a Dold manifold. In "Erzeugende der Thomschen Algebra $\mathfrak N$", Dold computes the mod 2 cohomology, Steenrod module structure, and Stiefel-Whitney classes of $P(m,n)$. The calculations are fairly straightforward, and I found them to be a good example when I was learning Stiefel-Whitney classes and wanted more examples than projective spaces.
The paper is in German, but it's nonetheless quite readable, especially with a dictionary.