Cells vary in their dynamic response to external stimuli, due to stochastic fluctuations and non-uniform progression through the cell cycle. Hence, single-cell studies are required to reveal the range of heterogeneity in their responses to defined perturbations, which provides detailed insight into signaling processes. Here, we present a time-lapse study using arrays of micro-trenches to monitor the timing of cell division and apoptosis in non-adherent cells at the single-cell level. By employing automated cell tracking and division detection, we precisely determine cell cycle duration and sister-cell correlations for hundreds of individual cells in parallel. As a model application we study the response of leukemia cells to the chemostatic drug vincristine as a function of cell cycle phase. The time-to-death after drug addition is found to depend both on drug concentration and cell cycle phase. The resulting timing and dose-response distributions were reproduced in control experiments using synchronized cell populations. Interestingly, in non-synchronized cells, the time-to-death intervals for sister cells appear to be correlated. Our study demonstrates the practical benefits of micro-trench arrays as a platform for high-throughput, single-cell time-lapse studies on cell cycle dependence, correlations and cell fate decisions in general.