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 $M=S^m, \mathbb{R}P^m,\mathbb{C}P^m, \mathbb{H}P^m$?



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). 


