  We prove that there are at most $(24-r_0)$ low-degree rational curves on
high-degree models of K3 surfaces with at most Du Val singularities, where
$r_0$ is the number of exceptional divisors on the minimal resolution. We also
provide several existence results in the above setting (i.e. for rational
curves on quasi-polarized K3 surfaces), which imply that for various values of
$r_0$ our bound cannot be improved.
