There are a number of arguments in both works by Wilde that have been socialized into public consciousness today. In An Ideal Husband, Wilde bravely makes the inequality between men and women in Victorian society explicit, rather than dwelling on coded or implicit arguments typical of most progressives of his time. For example, he states outright that a man's life is of more value than a woman's.
Wilde also comments on how language, rather than being a neutral vehicle for expression, often forms rhetorical games that are used to squelch the agency of oppressed groups. For example, when Lord Caversham appeals to the phrase "common sense" to argue that women are less functional and independent, he ironically only succeeds in making a self-referential statement about his own ignorance. These insights are useful tools for understanding how to identify language that oppresses rather than illuminates; for example, they can be used to discriminate between fake news and rigorous reporting when traversing large amounts of digital content.
In Lady Windermere's Fan, Wilde depicts the elite as a group that hangs onto the old, tired symbols of class difference while trying to come off as intellectual and relevant. This mismatch between one's intent and how one is actually perceived mirrors how public identities, especially in the age of social media, are mostly constructed in order for one to pass as a category of subject, obfuscating a poverty of content.
Many themes from An Ideal Husband and Lady Windermere's Fan are quite usable today. One of the most salient themes is the preservation of social reputation. In both plays, you can feel the preoccupation with being, or remaining, a part of a higher social stratum and the need to appear to be successful, "proper," and well-liked.
Mrs. Erlynne, a social outcast, wants desperately to get back in a social set in Lady Windermere's Fan. Lord Windermere's fear of his wife finding out that Erlynne is her mother is significant because he knows that it would stain their reputation, as they are part of an elevated social circle.
Sir Robert Chiltern built the reputation of "an ideal man" in An Ideal Husband knowing very well that it was a lie. He made his fortune because he once was an inside trader and a seller of state secrets. Being "found out" would bring ruin to his social standing. This is the reason why he has to stop the shady Mrs. Cheveley from telling the newspapers what she knows about him.
Another topic in the plays that still resonate to this day is infidelity. Mrs. Erlynne, Lady Windermere, and Lady Chiltern break their marital vows and they, respectively, elope, threaten to elope with, or plan to go seek refuge in, another man. This seems to happen quite quickly at the onset of difficulty. There is no attitude of "through better or worse" for these ladies.
Therefore, keeping up appearances, staying true to a moral compass, and marital infidelity are some of the issues in the play that are still relevant in our modern society.