Single-cell CRISPR screens (perturb-seq) link genetic perturbations to phenotypic changes in individual cells. The most fundamental task in perturb-seq analysis is to test for association between a perturbation and a count outcome, such as gene expression. We conduct the first-ever comprehensive benchmarking study of association testing methods for low multiplicity-of-infection (MOI) perturb-seq data, finding that existing methods produce excess false positives. We conduct an extensive empirical investigation of the data, identifying three core analysis challenges: sparsity, confounding, and model misspecification. Finally, we develop an association testing method - SCEPTRE low-MOI - that resolves these analysis challenges and demonstrates improved calibration and power.