Akaky's meeting with the "certain important person" (situational irony)
“It was to this important person that our Akaky Akakievich came, and came at a most unfavorable moment, very inopportune for himself, though very opportune for the important person” (p. 416).
The narrator flags for the reader this situational irony: Akaky comes to the important person with a request to assist in the investigation of the theft of his overcoat. Unfortunately for Akaky, this is a bad time for him to come, as the important person has a guest who he will want to impress by treating Akaky with imperious cruelty. Hence there is a stated irony in the fact that Akaky's goals are directly contradicted by the important person's goals, dooming Akaky to failure.
Writers making fun of titular councillors (verbal irony)
“... he was what is called an eternal titular councillor, at whom, as is known, all sorts of writers have abundantly sneered and jeered, having the praiseworthy custom of exerting themselves against those who can’t bite.” (p. 394-5)
The narrator here employs a kind of verbal irony, describing the habit of the writer of making fun of titular councillors as a “praiseworthy custom.” By juxtaposing something that is not very nice—making fun of people who cannot defend themselves—with sarcastic praise, the narrator indicates his disapproval of this custom.
The hostile weather in St. Petersburg (verbal irony)
“Owing to the generous assistance of the Petersburg climate, the illness developed more quickly than might have been expected...”(p. 418)
This is an example of verbal irony because by “generous,” the narrator is observing how the hostile cold weather of St. Petersburg accelerated the path of Akaky towards death.
The police order to catch Akaky (verbal irony)
“An order was issued for the police to catch the dead man at all costs, dead or alive…” (p. 420)
This is an example of verbal irony because the fact that Akaky is dead but the police wish to catch his ghost renders the cliché “to catch someone dead or alive” absurd.