If you read the definition on Wikipedia, you'll see that they allow characteristic classes to be defined on general principal $G$-bundles (vector bundles being subsumed in this general case by looking at the frame bundle (or subbundle thereof)). However, all of the 'big name' characteristic classes I am aware of (Stiefel-Whitney, Chern, Pontryagin, Euler, Todd) are defined in the special case of vector bundles (possibly with extra structure).
Are there any 'naturally occurring' examples of characteristic classes defined on principal $G$-bundles that are not frame bundles? (I say 'naturally occurring' because it is easy enough to cook up an example using the Chern-Weil Theorem, but I would prefer the example to actually be useful in practice.)