Spread mooring systems (SMS) may experience large-amplitude oscillations in the horizontal plane due to slow-drift loads. In the literature, this phenomenon is attributed to resonance. In this paper, it is shown that this conclusion is only partially correct. This phenomenon is investigated using nonlinear stability and bifurcation analyses which reveal an enhanced picture of the nonlinear dynamics of SMS. Catastrophe sets are developed in a parametric design space to define regions of qualitatively different system dynamics for autonomous SMS, including mean drift forces. Limited time simulations are performed to verify the qualitative conclusions drawn on the nonlinear dynamics of SMS using catastrophe sets. Slowly varying drift forces are studied as an additional excitation on the autonomous SMS and simulations reveal that slow drift may cause resonance or bifurcations with stabilizing or destabilizing morphogeneses. The mathematical model of SMS is based on the slow-motion maneuvering equations in the horizontal plane (surge, sway, yaw), including hydrodynamic forces with terms up to third-order, nonlinear restoring forces from mooring lines, and environmental loads due to current, wind, and wave-drift.