How does Oscar Wilde explore gender identity in his society comedies?
Each of these comedies explore gender identity in a subtle way, particularly regarding males, since Wilde was quite aware of the Criminal Law Amendment Act of 1885, also known as the Labouchere Amendment. This particular amendment attacked "gross indecency" acts. These acts were not specifically depicted but, later on, they referred to homosexual acts among males; not females: Queen Victoria insisted that "women did not do those things". Moving on...
Oscar Wilde gave a good understanding of gender identity in both males and women. This answer will focus of gender identity in women, considering that Victorian society ostracized and limited them incredibly. Wilde noted this.
In The Importance of Being Earnest gender identity is found in the way in which the females serve as foils to the male characters; they have the same confused identities, the same indecision, and the same wonder about what is awaiting for them in a near future. When Wilde "pairs them up", he does it with the intention of making love seem ridiculous, and courtship to seem worthless. Hence, the play has the subtitle of being "A trivial comedy for serious people".
In Lady Windermere's Fan gender identity is more evident because we find a husband FORCING his wife to accept the entrance of a strange woman into their society, and he would not even explain why. Poor Lady Windermere had to put up with this, despite of her ardent disagreement and, in the end, she had to accept this strange woman in her home. This is Wilde's way of criticizing the sanctimonious society that accused HIM of being immoral. Married men were even worse with their wives.
A Woman of No Importance and An Ideal Husband show women who are not scared of stating their wants and their needs. They defy the ideal Victorian prudish women, and they also challenge the men of their male-dominated society into, either, make a "deal" with them, or blackmailing the males altogether. These two plays make it evident that Wilde was quite aware of the dominance of males and that he was more than willing to place women in a different (and more powerful) role.