How to recognize a vector bundle? Given a connected topological space $E$, under which conditions is it possible to find a subspace $B$ such that $E$ can be regarded as a (rank $n$) vector bundle over $B$?
Is it possible to find the conditions and the $B$'s if one moves to the more rigid differentiable, holomorphic or algebraic setting?
What if we restrict to the case: dim $E$ = 2, dim $B$ = 1? When is a surface the total space of a line bundle?
 A: The total space of a vector bundle is homotopy-equivalent to the base. Hence, for example,
the only connected surfaces which are total spaces of real line bundles are the plane, the annulus and the Möbius strip.
A: In smooth manifolds, Grabowski and Rotkiewicz - Higher vector bundles and multi-graded symplectic manifolds has a condition for when a monoid action $(\mathbb{R}^+, \cdot, 1)$ on a manifold $E$ induces a vector bundle structure where $E$ is the total space. I have a similar result in a recent paper (Vector bundles and differential bundles in the category of smooth manifolds), so that a morphism $\lambda:E \to TE$ induces a vector bundle where $E$ is the total space whenever $\lambda$ satisfies some coherences and a certain pullback diagram (these are called differential bundles in a tangent category). The total space is obtained by splitting the idempotent $p \circ \lambda:E \to E$, where $p$ is the tangent projection (it's a consequence of the coherences on $\lambda$ that $p\circ \lambda$ is an idempotent).
I don't know of any similar results that hold for general topological vector bundles, but I would be interested in seeing them!
