A thin "rope" of viscous fluid falls from a sufficient height of coils as it approaches a rigid surface. Here we perform a linear stability analysis of steady coiling, with particular attention to the "inertio-gravitational" regime in which multiple states with different frequencies exist at a fixed fall height. The basic states analyzed are numerical solutions of asymptotic "thin-rope" equations that describe steady coiling. To analyze their stability, we first derive in detail a set of more general equations for the arbitrary time-dependent motion of a thin viscous rope. Linearization of these equations about the steady coiling solutions yields a boundary-eigenvalue problem of order 21 which we solve numerically to determine the complex growth rate. The multivalued portion of the curve of steady coiling frequency vs height comprises alternating stable and unstable segments whose distribution agrees closely with high-resolution laboratory experiments. The dominant balance of (perturbation) forces in the instability is between gravity and the viscous resistance to bending of the rope; inertia is not essential, although it significantly influences the growth rate.