This short paper reviews the main contributions of our recent work on static analysis of existential rules (a.k.a. tuple-generating dependencies). Between such rules, several kinds of logical relationships - also called dependencies in an unfortunate clash of terminology - are of interest, but their computation highly intractable (Σ^P₂-complete). We develop new, optimised procedures for this task, and present a prototype implementation that scales to rule sets with more than 100,000 rules. This allows us to perform much faster acyclicity checks and to identify rule sets that admit efficient core computation via the standard chase.