  We study the Bridgeland stability of line bundles on surfaces using
Bridgeland stability conditions determined by divisors. We show that given a
smooth projective surface $S$, a line bundle $L$ is always Bridgeland stable
for those stability conditions if there are no curves $C\subseteq S$ of
negative self-intersection. When a curve $C$ of negative self-intersection is
present, $L$ is destabilized by $L(-C)$ for some stability conditions. We
conjecture that line bundles of the form $L(-C)$ are the only objects that can
destabilize $L$, and that torsion sheaves of the form $L(C)|_C$ are the only
objects that can destabilize $L[1]$. We prove our conjecture in several cases,
and in particular for Hirzebruch surfaces.
