Oscar Wilde's play The Importance of Being Earnest certainly qualifies as a comedy of manners. A comedy of manners is a work that humorously satirizes the manners of a particular social group, typically the upper class (upper-class problems tend to have more potential for humor than lower-class problems: for example, a woman in the upper class might spend hours trying to figure out what to wear to an important party while a lower-class woman might spend hours trying to figure out what to feed her hungry children). Classifying the work as a satire means that the text would not only point out flaws in this group but would also suggest that changes should be made to reduce or eliminate those flaws. In the play, we definitely see Wilde poking fun at the upper class, and especially their frivolity and obsession with appearances.
However, the play cannot be characterized as a melodrama because -- though it has exaggerated characters (and some exciting events) -- it is not intended to appeal to the audience's emotions. We are supposed to laugh at these characters and the absurd situations they create for themselves; we are not supposed to be affected in some highly emotional way. Consider the scene where Jack and Algernon are eating muffins; we laugh at the sheer ridiculousness of the situation: they fight over who has eaten more muffins and whether or not it is appropriate to eat muffins when one is upset. Such scenes are absolutely designed to point out the absurdity and frivolity of upper-class problems in a humorous way, not to force the reader into some deep feelings about muffin-eating propriety.