  We compare spaces of non-singular algebraic sections of ample vector bundles
to spaces of continuous sections of jet bundles. Under some conditions, we
provide an isomorphism in homology in a range of degrees growing with the jet
ampleness. As an application, when $\mathcal{L}$ is a very ample line bundle on
a smooth projective complex variety, we prove that the rational cohomology of
the space of non-singular algebraic sections of $\mathcal{L}^{\otimes d}$
stabilises as $d \to \infty$ and compute the stable cohomology. We also prove
that the integral homology does not stabilise using tools from stable homotopy
theory.
