Many efforts have been dedicated to identifying restrictions on ontologies
expressed as tuple-generating dependencies (tgds), a.k.a. existential rules,
that lead to the decidability of answering ontology-mediated queries (OMQs).
This has given rise to three families of formalisms: guarded, non-recursive,
and sticky sets of tgds. We study the containment problem for OMQs expressed in
such formalisms, which is a key ingredient for solving static analysis tasks
associated with them. Our main contribution is the development of specially
tailored techniques for OMQ containment under the classes of tgds stated above.
This enables us to obtain sharp complexity bounds for the problems at hand.