Recent advancements in functional genomics have provided an unprecedented ability to measure diverse molecular modalities, but predicting causal regulatory relationships from observational data remains challenging. Here, we leverage pooled genetic screens and single-cell sequencing (Perturb-seq) to systematically identify the targets of signalling regulators in diverse biological contexts. We demonstrate how Perturb-seq is compatible with recent and commercially available advances in combinatorial indexing and next-generation sequencing, and perform more than 1,500 perturbations split across six cell lines and five biological signalling contexts. We introduce an improved computational framework (Mixscale) to address cellular variation in perturbation efficiency, alongside optimized statistical methods to learn differentially expressed gene lists and conserved molecular signatures. Finally, we demonstrate how our Perturb-seq derived gene lists can be used to precisely infer changes in signalling pathway activation for in vivo and in situ samples. Our work enhances our understanding of signalling regulators and their targets, and lays a computational framework towards the data-driven inference of an 'atlas' of perturbation signatures.