  We prove new results on single point Seshadri constants for ample line
bundles on hyperelliptic surfaces. Given a hyperelliptic surface $X$ and an
ample line bundle $L$ on $X$, we show that the least Seshadri constant
$\varepsilon(L)$ of $L$ is a rational number when $X$ is not of type 6. We also
prove new lower bounds for the Seshadri constant $\varepsilon(L,1)$ of $L$ at a
very general point.
