How does Hawthorne use irony in "Young Goodman Brown"?
All three types of irony are present. In fact, sometimes all three are present in a very scene.
An example is early in the story when Brown fears someone might see him walking and conversing with the dark stranger. Of course, he spies a woman hurrying toward them. Brown notices that it is Goody Cloyse, whom he states is his is his spiritual advisor. Here Hawthorne is setting us up for all kinds of irony.
First, situational irony - what do we expect to happen (and maybe if you have read a lot of Hawthorne's work, it really isn't ironic at all)? The first time I read it, I didn't expect this pious Salem woman, who also taught Brown his catechism, to be a witch.
Next, verbal irony - Brown has already admitted that Cloyse is a very important person to him, but Cloyse admits to the devil that Brown is really a silly fellow, so Brown hears the exact opposite of what he expects from his former catechism teacher and his current spiritual advisor.
Finally, dramatic irony - from Cloyse we learn that there is a new lad to be welcomed into the community of witches and sinners via the black mass. This comment seems to be lost on Brown, but the reader realizes that the new lad is likely going to be none other than Brown himself.
These are just a few examples of the irony used by Hawthorne. There are certainly others - the climax of the story and the resolution are also great examples.