He clearly has a sense of sight, he responds to auditory stimuli so he has a sense of hearing, but does he ever react to a smell? Does his beheading being the method of death affect any of his senses?
I shall answer for ghosts in general, as there is no reason to believe that any one ghost's ability to smell is different from another's.
The one example of ghosts detecting odors that comes to mind is at Nick's deathday party:
Harry watched, amazed, as a portly ghost approached the table, crouched low, and walked through it, his mouth held wide so that it passed through one of the stinking salmon. "Can you taste it if you walk through it?" Harry asked him. "Almost," said the ghost sadly, and he drifted away. "I expect they've let it rot to give it a stronger flavor," said Hermione knowledgeably, pinching her nose and leaning closer to look at the putrid haggis. (from Chamber of Secrets, Chapter 8)
This does not give us a definitive answer, but it would seem to imply that ghosts do lack the sense of taste, yet are able to simulate that sense by appealing to their sense of smell instead. This is logical, as smell and taste are closely related senses.
Thus it would seem that ghosts do have the ability to smell. As the ghosts do not seem to react as aversely to the aroma as the trio does, however, it may be that the sense is somewhat subdued compared to the sense of smell in a human — or perhaps they are simply so desperate for another taste of food that they are more willing to tolerate the smell.
The Pottermore article on Ghosts says that they can not experience physical pleasure:
Having chosen a feeble simulacrum of mortal life, ghosts are limited in what they can experience. No physical pleasure remains to them
Although the Deathday Party in Book two does kind of imply that Ghosts can somewhat experience smell.
I expect they've let it rot to give it a stronger flavor.