If $M$ is paracompact (let us assume smooth manifold of dimension $d$), then one has that the set of isomorphism classes of vector bundles of rank $n$ over $M$ is isomorphic to the set of homotopy classes $V_n:=[M,G_n\left(\mathbb{R}^{n+d}\right)]$ from $M$ to a Grassmannian. This holds both in the topological and the smooth category over $M$.
I was wondering if there is any "nice" topological/smooth structure on $V_n$?
Does it carry a well-behaved and non-trivial topology?
Are there results on how "many" elements $V_n$ usually has? Finitely many, Countably many?
This is really not my area, so these questions may have simple, well-known answers.