  We classify all complex surfaces with quotient singularities that do not
contain any smooth rational curves, under the assumption that the canonical
divisor of the surface is not pseudo-effective. As a corollary we show that if
$X$ is a log del Pezzo surface such that for every closed point $p\in X$, there
is a smooth curve (locally analytically) passing through $p$, then $X$ contains
at least one smooth rational curve.
