Cellular biomechanics play a pivotal role in the pathophysiology of several diseases. Unfortunately, current methods to measure biomechanical properties are invasive and mostly limited to the surface of a cell. As a result, the mechanical behaviour of subcellular structures and organelles remains poorly characterised. Here, we show three-dimensional biomechanical images of single cells obtained with non-invasive, non-destructive Brillouin microscopy with an unprecedented spatial resolution. Our results quantify the longitudinal elastic modulus of subcellular structures. In particular, we found the nucleoli to be stiffer than both the nuclear envelope (p < 0.0001) and the surrounding cytoplasm (p < 0.0001). Moreover, we demonstrate the mechanical response of cells to Latrunculin-A, a drug that reduces cell stiffness by preventing cytoskeletal assembly. Our technique can therefore generate valuable insights into cellular biomechanics and its role in pathophysiology.