Audiences continue to be fascinated and relate with Katherina because of the struggles she faces with not fitting into a very concise, social mould. There's no denying that The Taming of the Shrew is patriarchy at its worst. Shakespeare presents to the audience Katherina - an intelligent, highly stubborn woman who is willing to challenge the sexist patriarchal ideologies of the 16th century. She is subjected to many things like verbal abuse, injustice and double standards, which all resonate with the audience.
Katherina doesn't fit into the social convention because she challenges the 'proper' gender roles of the 16th century. We see Katherina challenging gender conventions when she protests against being treated like a commodity in terms of marriage which at the time was like a business or financial transaction supposed to benefit the families involved. Katherina makes it clear at the beginning of the play that she does not wish to be married to someone merely because of her dowry, much to Bianca's despair. In Act 2 Scene 1 Baptista agrees for Petruchio to marry her. Katherina is very against the idea, twisting his words, insulting and even striking Petruchio. She is finally shocked to silence when even despite her best efforts to protest, Petruchio deceives Baptista in claiming that Katherina wants to be married to him and she "hung about my neck, and kiss on kiss She vied so fast". Katherina also challenges gender roles in that she will willingly stand up for herself, which was very unlike any woman of the time. This did not help her case in being a 'shrew' with a loud, ill-tempered personality and a "razor-sharp tongue". We see this in Act 1 Scene 1 when Katherina is being openly insulted and abused by Bianca's suitors and refuses to step down and accept the fact she is being called a "devil" and a "wench". She retaliates with speech that is witty and shreds the suitor's comments,