  Our main point of focus is the set of closed geodesics on hyperbolic
surfaces. For any fixed integer $k$, we are interested in the set of all closed
geodesics with at least $k$ (but possibly more) self-intersections. Among
these, we consider those of minimal length and investigate their
self-intersection numbers. We prove that their intersection numbers are upper
bounded by a universal linear function in $k$ (which holds for any hyperbolic
surface). Moreover, in the presence of cusps, we get bounds which imply that
the self-intersection numbers behave asymptotically like $k$ for growing $k$.
