Biological systems at various levels of organisation exhibit robustness, as well as phenotypic variability or evolvability, the ability to evolve novel phenotypes. We still know very little about the relationship between robustness and phenotypic variability at levels of organisation beyond individual macromolecules, and especially for signalling circuits. Here, we examine multiple alternate topologies of the Saccharomyces cerevisiae target-of-rapamycin (TOR) signalling circuit, in order to understand the circuit’s robustness and phenotypic variability. We consider each of the topological variants a genotype, a set of alternative interactions between TOR circuit components. Two genotypes are neighbours in genotype space if they can be reached from each other by a single small genetic change. Each genotype (topology) has a signalling phenotype, which we define via the concentration trajectories of key signalling molecules. We find that the circuits we study can produce almost 300 different phenotypes. The number of genotypes with a given phenotype varies very widely among these phenotypes. Some phenotypes have few associated genotypes. Others have many genotypes that form genotype networks extending far through genotype space. A minority of phenotypes accounts for the vast majority of genotypes. Importantly, we find that these phenotypes tend to have large genotype networks, greater robustness and a greater ability to produce novel phenotypes. Thus, over a broad range of phenotypic robustness, robustness facilitates phenotypic variability in our study system. Our observations show parallels to studies on macromolecules, suggesting that similar principles might govern robustness and phenotypic variability in biological systems. Our approach points a way towards mapping genotype spaces in complex circuitry, and it exposes some challenges such mapping faces.