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We investigate the trajectories of rigid fibres as they are transported in a pressuredriven flow, at low Reynolds number, in shallow Hele-Shaw cells. The transverse confinement and the resulting viscous friction on these elongated objects, as well as the lateral confinement (i.e. the presence of lateral walls), lead to complex fibre trajectories that we characterize with a combination of microfluidic experiments and simulations using modified Brinkman equations. We show that the transported fibre behaves as an oscillator for which we obtain and analyse a complete state diagram. [ABSTRACT FROM AUTHOR]