Scottie following Madeleine (Dramatic Irony)
As Scottie begins working as a private detective for Gavin Elster, following Madeleine around, we know that he is following her, but Madeleine does not. Thus, a kind of dramatic irony is at play. This will eventually become unraveled by the realization, later in the film, that Madeleine actually did know that Scottie was following her and was simply playing a part, but in the beginning of his pursuit, the viewer seems to know something she does not.
Scottie follows Madeleine back to his house (Situational Irony)
The day after saving Madeleine from drowning in the San Francisco Bay, Scottie follows her in his car at a distance to spy on her whereabouts. Ironically enough, however, she ends up leading him all the way back to his house, where she is going to deliver a thank-you note.
Judy is Madeleine (Dramatic Irony)
After Scottie meets Judy Barton, he feels sure that she is somehow a manifestation of the dead Madeleine, even though this is completely impossible. When he leaves her hotel room, we see Judy take out a pen and paper and write a confession revealing that she is indeed "Madeleine," but that she was hired by Gavin Elster to pose as his wife in order to cover up his murder of his actual wife, whose body actually fell from the bell tower. In this moment, we know that Judy is "Madeleine," but Scottie has no idea, resulting in some very tense dramatic irony that extends into the beginning stages of Scottie and Judy's courtship.
Judy falls to her death (Situational Irony)
At the end of the film, Scottie discovers that Judy is actually "Madeleine," and, out of his sense of betrayal, he brings her to the bell tower at the mission in order to reveal to her that he knows her true identity. He rants and raves at her, and she begs for his forgiveness, falling into his arms and asking for him to show her love. It is an ambiguous moment of reconciliation, and we are not sure if Scottie will take her back, but before we can find out, Judy is startled by the approach of a nun who has come to check on what is going on. Judy is so alarmed that she falls to her death. Ironically enough, she dies in the same way that she was alleged to have died in the Gavin Elster plot.