Is there anywhere we use a fibration which is not a fiber bundle What I currently meet are all fiber bundles.
 A: There's a canonical example of fibrations used in algebraic topology which are not fibre bundles.  That is, given a continuous function $f : X \to Y$ there is a homotopy-equivalence $\phi : X' \to X$ and a fibration $f' : X' \to Y$ such that $f \circ \phi$ is homotopic to $f'$.  I believe this idea goes back to Serre (or perhaps earlier).  The fibre of $f'$ is called the homotopy-fibre of $f$.  
A common usage of this construction is with the Postnikov Tower of a space.  In his dissertation Jean-Pierre Serre used this (and some "closure" observations of the Serre spectral sequence of a fibration, then it was called Serre $\mathcal C$-theory, nowadays this technology is subsumed in localization) to show that most of the homotopy-groups of the spheres are finite.  
A: I think that most examples of fibrations that you can think of off the top of your head are going to be fiber bundles (although this is probably only because of my limited experience).  On the other hand, it's much easier to just prove that things are fibrations, because the definition is much weaker.  As far as homotopy theory is concerned, this is going to be the same thing.
