Recently I've been reading the Jankins/Neumann notes Lectures on Seifert Manifolds (PDF). In the notes it is claimed that Seifert spaces are almost fiber bundles (with $S^1$ fiber), except around their exceptional fibers.
The example given is of taking a a solid cylinder $D^2\times I$, and gluing the ends with a rational $\frac{p}{q}$ turn; then the central fiber $0\times I$ becomes a circle in the resulting space that doesn't have a locally trivial neighborhood. But this space is still a solid torus, so of course it still is a fiber bundle $S^1\rightarrow M\rightarrow D^2$.
I think the point is that it isn't a fiber bundle in the "obvious" way, but I'm now curious about finding a connected Seifert fiber space $M$ that is not a fiber bundle $S^1\rightarrow M\rightarrow F$ for some surface $F$. Does such an $M$ exist? Ideally $M$ would have as many of the following qualifiers as possible: compact; orientable; without boundary.
Definitely in many cases there is no actual fiber bundle structure. In fact, in most cases the Seifert fibration on a closed orientable Seifert fibered 3-manifold is unique, so if it isn't already a fiber bundle then it doesn't have an alternate structure as an honest $S^1$-bundle over a surface. For a precise statement of this uniqueness that lists all the exceptions, see Theorem 2.3 of Hatcher's notes on 3-manifolds, which can be found here.