Instance segmentation of single cells from microscopy images is critical to quantitative analysis of their spatial and morphological features for many important biomedical applications, such as disease diagnosis and drug screening. However, the high densities, tight contacts, and weak boundaries of the cells pose substantial technical challenges. To overcome these challenges, we have developed a new instance segmentation model, which we refer to as single-cell Transformer segmenter (SCTS). It utilizes a Swin Transformer as its backbone, combining the global modeling capabilities of a Transformer and the local modeling capabilities of a convolutional neural network (CNN) to ensure model adaptability to different cell sizes, shapes, and textures. It also embeds a three-class (background, cell interior, and cell boundary) semantic segmentation branch to classify pixels and to provide semantic features for downstream tasks. The prediction of boundary semantics improves boundary awareness, and the differentiation between foreground and background semantics improves segmentation integrity in regions with weak signals. To reduce the need for annotated training data, we have developed an augmentation strategy that randomly fills instances of single cells into open spaces of training images. Experiments show that our model outperforms several state-of-the-art models on the LIVECell dataset and an in-house dataset. The code and dataset of this work are openly accessible at https://github.com/cbmi-group/SCTS.