We describe the trapping and release of giant unilamellar vesicles (GUVs) in a thin and wide microfluidic channel, as they cross indentations etched in the channel ceiling. This trapping results from the reduction of the membrane elastic energy, which is stored in the GUV as it squeezes to enter into the thin channel. We demonstrate that GUVs whose diameter is slightly larger than the channel height can be trapped and that they can be untrapped by flowing the outer fluid beyond a critical velocity. GUVs smaller than the channel height flow undisturbed while those much larger cannot squeeze into the thin regions. Within the range that allows trapping, larger GUVs are anchored more strongly than smaller GUVs. The ability to trap vesicles provides optical access to the GUVs for extended periods of time; this allows the observation of recirculation flows on the surface of the GUVs, in the forward direction near the mid-plane of the channel and in the reverse direction elsewhere. We also obtain the shape of GUVs under different flow conditions through confocal microscopy. This geometric information is used to derive a mechanical model of the force balance that equates the viscous effects from the outer flow with the elastic effects based on the variation of the membrane stretching energy. This model yields good agreement with the experimental data when values of the stretching moduli are taken from the scientific literature. This microfluidic approach provides a new way of storing a large number of GUVs at specific locations, with or without the presence of an outer flow. As such, it constitutes a high-throughput alternative to micropipette manipulation of individual GUVs for chemical or biological applications.