  Let $X$ be a complex nonsingular projective surface and let $L$ be an ample
line bundle on $X$. We study multi-point Seshadri constants of $L$ at singular
points of certain arrangements of curves on $X$. We pose some questions about
such Seshadri constants and prove some results in the case of star arrangements
of curves. We also study the configurational Seshadri constants for curve
arrangements on surfaces and compare them with the usual Seshadri constants. We
give several examples illustrating the properties that we study.
