The problem of multiple equilibria in steady towing of a floating body is considered. The two coordinates of the towing point are the main bifurcation parameters. An approach to bifurcation of steady-state equilibria using singularity theory reveals all qualitatively different bifurcation diagrams that occur locally. It is shown that these bifurcation problems may be viewed as paths in the universal unfolding space of the cusp catastrophe. The organizing centre for the towing problem is the pitchfork singularity. Numerical calculations suggest that results obtained by singularity-theory techniques are valid globally.