One of the unresolved questions about 3-manifolds is the generalized Smale conjecture, which roughly interpreted asks for the homotopy type of the space of diffeomorphisms of a 3-manifold. Smale originally conjectured that $Diff(S^3)\simeq O(4)$, and this was proven by Hatcher. He also worked out the homotopy type of diffeomorphisms of Haken 3-manifolds. Another interpretation of Smale's question is that the space of round (constant sectional curvature $=1$) metrics on $S^3$ is contractible. Gabai proved the analogous statement that the space of hyperbolic metrics on a hyperbolic 3-manifold is contractible, and recently McCullough and Soma have dealt many small (non-Haken) Seifert-fibered spaces. However, the case of the generalized Smale conjecture for elliptic manifolds is still open (see however the work of Hong et. al.). I think this is an important open question, and it would be useful to have a unified proof of these results (in particular, Gabai's results makes use of a computer-aided proof of the existence of "non-coalescable insulator families").
One possible approach is to try to prove that the space of metrics is contractible (on a constant curvature manifold) by showing that all the homotopy groups vanish (it is known to be of the homotopy type of a CW-complex, so this suffices). This was the approach that Gabai took. You can fill in a sphere of constant curvature metrics with a ball of Riemannian metrics, since the space of Riemannian metrics is convex. Then you could try to "flow" towards a ball of constant curvature metrics using Ricci flow (which would stay fixed on the boundary of the ball). The issue is that under Ricci flow, singularities may occur. However, what I hope is that some sort of canonical Ricci-flow with surgery may be used to fill in the sphere with a ball of constant curvature metrics. Thus, I see it as an important question for 3-manifold topology to obtain an understanding of a version of Ricci flow flow-with-surgery and Perelman's proof of geometrization for families of Riemannian metrics. This approach for more general Seifert fibered spaces would be trickier, since one would probably have to get a very good idea of how the collapsing occurs at infinite time under Ricci flow, and prove finiteness of surgeries.