Let (Formula presented.) be integers and let (Formula presented.) denote the Hirzebruch surface with invariant (Formula presented.). We compute the Seshadri constants of an ample line bundle at an arbitrary point of the (Formula presented.) -point blow-up of (Formula presented.) when (Formula presented.) and at a very general point when (Formula presented.) or (Formula presented.). We also discuss several conjectures on linear systems of curves on the blow-up of (Formula presented.) at (Formula presented.) very general points.