at.algebraic topology - triviality of fibre bundles

are there some general method in judging if a fible bundle is trivial?

At least,for vector bundles,there is a well-developed theory,that is Charicteristic Classes.the triviality of vector bundles is equivalent to the vanishment of its characteristic classes.

for principal bundles,the triviality is equivalent to the exsience of a cross section.
(any good perspective on this assertion?besides,how to tell if there is a cross section)

for general fiber bundle (E,F,B,G),(here,E is total space,F the fiber,B the base space,G the structure group),we can construct its associate principal bundle (E',G,B,G),i.e.to replace the fiber F with the topological group G.there is a theorem that a fiber bundle is trivial iff its associate principal bundle is trivial.

hence the problem is reduced to find a cross section of principal bundle.

I want to know if there is some other methods that are more usable?
Thank you!