Abstract: Given a closed, Riemannian 3-manifold (N,g) without symmetries (more precisely: generic) and a non-negative integer p, can we say something about the number of minimal surfaces it contains whose Morse index is bounded by p? More realistically, can we prove that such number is necessarily finite? This is the classical "generic finiteness" problem, which has a rich history and exhibits interesting subtleties even in its basic counterpart concerning closed geodesics on surfaces. It is this question that we settle: indeed, we prove that when g is a bumpy metric of positive scalar curvature either finiteness holds or N does contain a copy of RP^3 in its prime decomposition, which is a sharp conclusion as we can exhibit specific obstructions to any further generalisation of such result. When g is assumed to be strongly bumpy (meaning that all closed, *immersed* minimal surfaces do not have Jacobi fields, a notion recently proved to be generic by White) then the finiteness conclusion is true for any compact 3-manifold without boundary.