Most mammalian genes have multiple polyA sites, representing a substantial source of transcript diversity regulated by the cleavage and polyadenylation (CPA) machinery. To better understand how these proteins govern polyA site choice, we introduce CPA-Perturb-seq, a multiplexed perturbation screen dataset of 42CPA regulators with a 3' scRNA-seq readout that enables transcriptome-wide inference of polyA site usage. We develop a framework to detect perturbation-dependent changes in polyadenylation and characterize modules of co-regulated polyA sites. We find groups of intronic polyA sites regulated by distinct components of the nuclear RNA life cycle, including elongation, splicing, termination, and surveillance. We train and validate a deep neural network (APARENT-Perturb) for tandem polyA site usage, delineating a cis-regulatory code that predicts perturbation response and reveals interactions between regulatory complexes. Our work highlights the potential for multiplexed single-cell perturbation screens to further our understanding of post-transcriptional regulation.