In single-cell RNA sequencing data, biological processes or technical factors may induce an overabundance of zero measurements. Existing probabilistic approaches to interpreting these data either model all genes as zero-inflated, or none. But the overabundance of zeros might be gene-specific. Hence, we propose the AutoZI model, which, for each gene, places a spike-and-slab prior on a mixture assignment between a negative binomial (NB) component and a zero-inflated negative binomial (ZINB) component. We approximate the posterior distribution under this model using variational inference, and employ Bayesian decision theory to decide whether each gene is zero-inflated. On simulated data, AutoZI outperforms the alternatives. On negative control data, AutoZI retrieves predictions consistent to a previous study on ERCC spike-ins and recovers similar results on control RNAs. Applied to several datasets and instances of the 10x Chromium protocol, AutoZI allows both biological and technical interpretations of zero-inflation. Finally, AutoZI's decisions on mouse embyronic stem-cells suggest that zero-inflation might be due to transcriptional bursting.