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.

9 pages, 10 figures