The Cat's cleanup (Situational Irony)
The Cat’s abrupt re-entrance into the house with a cleaning vehicle subverts our perception of the Cat as a mischief-making rebel who ignores rules. The shock of the Cat's gesture intensifies when he claims that he "always picks up all my [his] playthings" and rapidly transforms the disarrayed house back to its original, neat condition (179. Situational irony suffuses the Cat’s clean-up effort: the last outcome we expect is for the Cat, who incited the mess and chaos in the house in the first place, to be the character who ultimately restores order in the house. This "plot twist" enriches the characterization of the Cat: he celebrates fun and shenanigans, while also respecting certain domestic rules and taking responsibility for his mess. With this, Seuss suggests that chaos and order are two principles that can coexist within an individual.
Mother's return (Dramatic Irony)
When the children's mother returns, the Cat has vanished, along with all evidence of his shenanigans. Thus the reader is aware of something—that is, everything that happened while she was away—of which the mother remains completely ignorant, leading to dramatic irony. The final lines of the story, in which the boy asks the reader whether they would tell their mother about the adventures with the Cat, play on this strong element of dramatic irony.