Nice things that can be proved easily with characteristic classes A bit of context for this question: as a project for my master's degree my supervisor asked me to understand the construction of Milnor's exotic spheres. After learning the heavy material (I knew very little algebraic topology so learning about characteristic classes counted as "heavy" for me) the proof of existence of these spheres is surprisingly short and easy. 
I have finished early and I would like to go on and explore other ideas, ideally results which are not unreasonably difficult and preferably are  "surprising" i.e. the conclusion is something other than the classification of vector bundles over a thing. Any pointers? :)
 A: I like this example. The Spheres $S^{2n}$ cannot be complex manifolds unless $n=0,1,3$.
One proves that $TS^{2n}$ does not have the structure of a complex vector bundle in these cases. If $TS^{2n}$ were a complex vector bundle, then $c_{n}(TS^{2n}) = e (TS^{2n})$, so $c_n (TS^{2n})$ is twice a generator of $H^{2n} (S^{2n}; \mathbb{Z})$. 
Case 1: $n=2m$ even. Then $p_m (TS^{2n}) = \pm c_{2m} (TS^{2n} \otimes \mathbb{C}) = c_{2m} (TS^{2n} \oplus \overline{TS^{2n}})$. By the product formula for Chern classes, this is $= \pm c_{2m}(TS^{2n}) + c_{2m} \overline{TS^{2n}}) =\pm 2 c_{2m }(TS^{2n}) \neq 0$. This is a contradiction since $TS^{2n} \oplus \mathbb{R}$ is trivial.
Case 2: $n \geq 4$. This is more difficult and relies on the Bott periodicity theorem, one of whose corollaries states that a the top Chern class $c_n (V)$ of a complex  vector bundle $V\to S^{2n}$ is divisible by $(n-1)!$
A: Secondary characteristic classes allow us to see the large scale behaviour of leaves of foliations of 3-manifolds by surfaces; see
MR1040572 (91h:57015) Reviewed
Ghys, Étienne(F-ENSLY)
L'invariant de Godbillon-Vey. (French) [The Godbillon-Vey invariant]
Séminaire Bourbaki, Vol. 1988/89.
Astérisque No. 177-178 (1989), Exp. No. 706, 155–181.
57R30 (58F18)

A: In this blog post you'll find a computation of the cohomology ring of a hypersurface of degree $d$ in $\mathbb{CP}^3$ using characteristic classes. This turns out to be a weirdly good exercise in using characteristic classes: the computation invokes, in order, Euler classes, Chern classes, Pontryagin classes, Wu classes, and Stiefel-Whitney classes, and doing it is what made me comfortable with characteristic class computations. 
