Meta-analyses have become the gold standard for synthesizing evidence from multiple clinical trials, and they are especially useful when outcomes are rare or adverse since individual trials often lack sufficient power to detect a treatment effect. However, when zero events are observed in one or both treatment arms in a trial, commonly used meta-analysis methods can perform poorly. Continuity corrections (CCs), and numerical adjustments to the data to make computations feasible, have been proposed to ameliorate this issue. While the impact of various CCs on meta-analyses with rare events has been explored, how this impact varies based on the choice of pooling method and heterogeneity variance estimator is not widely understood. We compare several correction methods via a simulation study with a variety of commonly used meta-analysis methods. We consider how these method combinations impact important meta-analysis results, such as the estimated overall treatment effect, 95% confidence interval coverage, and Type I error rate. We also provide a website application of these results to aid researchers in selecting meta-analysis methods for rare-event data sets. Overall, no one-method combination can be consistently recommended, but some general trends are evident. For example, when there is no heterogeneity variance, we find that all pooling methods can perform well when paired with a specific correction method. Additionally, removing studies with zero events can work very well when there is no heterogeneity variance, while excluding single-zero studies results in poorer method performance when there is non-negligible heterogeneity variance and is not recommended.