The poor efficiency of microfluidic single cell trapping is currently restricting the full potential of state-of-the-art single cell analyses. Using fluid dynamics simulations in combination with particle image velocimetry to systematically optimize trap architectures, we present a microfluidic chip with enhanced single cell trapping and on-chip culture performance. Upon optimization of trap geometries, we measured trapping efficiencies of up to 97%. Our device also enables the stable, relatively long-term culture of individual non-adherent mammalian cells in high-throughput without a significant decrease in cell viability. As a first application of this platform we demonstrate the automated separation of the two daughter cells generated upon single cell division. The reliable trapping and re-trapping of mammalian cells should for example provide the fundament for novel types of investigations in stem cell and tumour cell biology, which depend on reliable tracking of genealogical relationships such as in stem cell lineage tracking.