What are the different types of irony used in the play? Where are they seen being used throughout the play?

Arsenic and Old Lace draws much of its comedy from irony, and it's chock full of multiple types—verbal, situational, and dramatic.
Situational irony occurs when events turn out the opposite of what characters or the audience expects. The whole premise of the play is ironic: that sweet, elderly spinsters could be responsible for murdering thirteen men. Other ironies are that a fiendish murderer, Jonathan, arrives and finds himself in competition with the sweet old ladies for being the most prolific murderer; that Teddy's harmless bugle playing is what draws the police to the home rather than the murders and the escaped convict; that Jonathan turns himself in when he mistakes Brophy's words on the telephone as a reference to himself rather than to O'Hara; and that Einstein has "disguised" Jonathan by giving him the face of a famous movie actor.
Verbal irony occurs when the words spoken or written convey a meaning opposite to their intended or received meaning. Near the beginning of the play, when the police officers are relating how sweet the aunts are, they say that the women don't really rent out their rooms. Brophy explains, "It's just their way of digging up people to do some good to." This turns out to be ironic because their advertisement for boarders is what lures unsuspecting men to their deaths. Later Elaine praises the women, saying, "Just look at your aunts—they're Brewsters, aren't they?—and the sanest, sweetest people I've ever known." By this time, the irony is clear because the audience already knows the women are serial killers.
Elaine's quote is also an example of dramatic irony—a type of irony that occurs when the audience or a character knows something that one or more characters do not. In perhaps one of the greatest scenes of dramatic irony to ever be played out on stage, Mortimer relates to Einstein the flawed plot of a play he has recently seen. As Mortimer re-enacts the stupidity of the play's main character, the audience sees Jonathan behind Mortimer acting out the cues as Mortimer narrates—actions that brought about the stupid main character's capture. Jonathan cuts the drapery cords and uses them to tie Mortimer. Einstein and the audience realize that Mortimer is narrating his own fate while he remains clueless.
Arsenic and Old Lace is replete with all kinds of irony, making the play utterly enjoyable and hilarious for the audience.