Through statistical analysis of datasets describing single cell shape following systematic gene depletion, we have found that the morphological landscapes explored by cells are composed of a small number of attractor states. We propose that the topology of these landscapes is in large part determined by cell-intrinsic factors, such as biophysical constraints on cytoskeletal organization, and reflects different stable signaling and/or transcriptional states. Cell-extrinsic factors act to determine how cells explore these landscapes, and the topology of the landscapes themselves. Informational stimuli primarily drive transitions between stable states by engaging signaling networks, while mechanical stimuli tune, or even radically alter, the topology of these landscapes. As environments fluctuate, the topology of morphological landscapes explored by cells dynamically adapts to these fluctuations. Finally we hypothesize how complex cellular and tissue morphologies can be generated from a limited number of simple cell shapes.