During Wilde's Victorian England, one of the most heated topics of conversation in the most fashionable drawing rooms of the time was the "problem of the East End". This particular issue is the most noticeable instance of class prejudice in the novel, for many good reasons.
Quick background: The East End slums of London, and the Whitechapel district were two sectors rife with disease, crime, poverty, hunger, and the most deplorable conditions ever to be present in an industrialized and rich nation such as England was under Queen Victoria. However, while the country grew as an economic force, the poor became even poorer, as more people moved into the city and less services were provided to tend to all of their needs. As a result, the upper classes would "be charitable" by showing off their financial status through philanthropy. To the question of what Lord Henry proposes to change about the East End situation in Chapter III, his answer was simply
I don't desire to change anything in England except the weather.
Notice how this aspect of Victorianism is ever-present in the conversation of the upper classes in The Picture of Dorian Gray. Consider also how the topic fades in discourse and disappears altogether when it is interrupted by either Lord Henry's epigrams, or by a more interesting topic. This is because, in reality, there is no interest whatsoever in the lower classes.
"You are really very comforting," warbled the duchess. "I have always felt rather guilty when I came to see your dear aunt, for I take no interest at all in the East End. For the future I shall be able to look her in the face without a blush."
In addition to the marked superiority shown by the upper classes through charity, class separation is also present in the events that led to the death of Dorian's mother. A beautiful woman, she ran away with a "penniless young fellow, a mere nobody, a subaltern in a foot regiment, or something of that kind." This "nobody" is actually Dorian's father. Due to this, it is said that Dorian's grandfather, Lord Kelso, sent a "Belgian Brute" to kill the man in a duel. Interestingly, although the matter was public business, it came to little consequence to Lord Kelso, who simply got a slight scorn but went on with his life. This is how insignificant the lower classes were to the rich.
The thing was hushed up, but, egad, Kelso ate his chop alone at the club for some time afterwards.
Also indicative of class separation is the relationship between Dorian and Sybil Vane. Sybil, who is an actress in a third rate theater in the slums represents the epitome of the lower class woman. Although respectable, actresses were often seen as lose women. However, she became for a little while Dorian's ideal of perfection. In turn, Sybil's devotion to Dorian was big enough to not even dare to ask his name that first night. All because he was an aristocrat and he had a "due" respect.
As a last note, within the realm of the theatre there were lots of social "undesirables", under the perspective of Wilde's England. The novel talks of "the greasy, greedy Jew" who managed the theater as well as of the "gypsies" who roamed the opium dens that Dorian entered to satisfy his gory tastes. Yet, as Wilde himself was not racially prejudiced, his mention of these descriptors denote that all prejudices are still connected with social status, first and foremost.