For my graduate (master) thesis I am studying the theory of Chern Classes. As a possible personal development the only sensible idea I have so far, and which I frankly think is impossible, is to work on the inverse problem, i.e., given a class in cohomology, is there a vector bundle which has that class as one of its Chern classes? The following question is to understand what it is already known about this and if maybe I might focus my attention on a small sub-problem and if it is an idea worth telling my supervisor at all or just forget about it.

1. Has this problem been analysed before? Is it a sensible matter to tackle?
2. I know how to find Chern Classes by taking conjugate invariant symmetric polynomials of the curvature matrix of a connection on the manifold (so de Rham Cohomology). Is this approach better, worse, equivalent to studying line divisors, picard groups and related? (Is there any other approach to finding Chern classes?)

EDIT

Ok, the problem seems rather involved with fundamental questions. Can you suggest one or more sub-problem which I might be able to work on in 3 months? Maybe even pre-existing results over which I can elaborate which explicit computations, examples, counter-examples, generalizations...