Recently there has been a renewed interest in using foamy suspensions of wood fibres as a carrier fluid in papermaking but there is a lack of fundamental understanding of the dynamics of such a three-phase system. In this article we propose a numerical model for the dynamics of an individual flexible fibre within a flowing foam, based on discrete-element methods. As is observed in a Newtonian shear flow, we observe that the fibre systematically experiences a tumbling instability: the disordered motion of bubbles cannot prevent the pseudo-periodical flip of the fibre. Our simulations show that the tumbling time decreases almost as the inverse of the strain rate. It also decays when the fibre length is increased, asymptoting to a finite constant. The tumbling time is found to be independent of the stiffness of the fibre. Because of their tumbling motion, long and flexible fibres spend most of the time in a coiled geometry. This would imply that using foam as a carrier fluid is not enough to keep fibres aligned with the flow. However, further refinements of the model will need to be considered to arrive at firm conclusions regarding alignment.