  In very anisotropic layered superconductors (e.g. Bi$_2$Sr$_2$CaCu$_2$O$_x$)
a tilted magnetic field can penetrate as two co-existing lattices of vortices
parallel and perpendicular to the layers. At low out-of-plane fields the
perpendicular vortices form a set of isolated vortex chains, which have
recently been observed in detail with scanning Hall-probe measurements. We
present calculations that show a very delicate stability of this isolated-chain
state. As the vortex density increases along the chain there is a first-order
transition to a buckled chain, and then the chain will expel vortices in a
continuous transition to a composite-chain state. At low densities there is an
instability towards clustering, due to a long-range attraction between the
vortices on the chain, and at very low densities it becomes energetically
favorable to form a tilted chain, which may explain the sudden disappearance of
vortices along the chains seen in recent experiments.
