In The Taming of the Shrew, did Katherina or Bianca have the more successful marriage?
The answer depends on the definition of "success," but we would probably say that Katherina has the most successful marriage. Both she and Petruchio are unorthodox characters who behave in unorthodox ways. By finally slipping into the "ideal wife" model, Katherina gives Petruchio little choice but to moderate his own eccentricity and live up to the "ideal husband" model of the time, which was not quite as tyrannical as some modern students might think (cf. Healey's article referenced below). Katherina's submission is a judo move, in which the strength of Petruchio's assault on her is turned against him. Now the eye of society will be directed towards him to see if he measures up.
Bianca, on the other hand, as shown by her behavior, has taken the superficially more dominant but actually more vulnerable role of the nagging, defiant wife. She can dominate her husband, but in an unsubtle way and one that is strongly depreciated by society. Thus, she is at first sight in a more empowered position, but in the long term she will attract much more hostility from society than Katherina will.
So, to sum up, in the short term it might seem that Bianca has made the better marriage (in the sense that she has more control and agency over her own life), but in the long term Katherina, if she continues to leverage social expectations in her favor, has a better chance of getting her way with the minimum of effort.
If things continue on the path they began, Katherine will have the more successful marriage. I might not have said so except for one telling moment in the last act.
Petruchio is obviously the "tamer" in their marriage, and Kate has learned the hard way how to get what she wants from him. When they are at her father's house, Kate is summoned to her husband. The gentlemen have placed a wager on which of their ladies will or will not come when beckoned. In those moments when Grumio is presumably fetching Kate, Petruchio is nervous; he's not sure she'll come. The arrogance he's shown in the taming process is not in evidence, and he shows a vulnerability which gives me hope that theirs will grow to be a more balanced partnership than the way it began. (And if I were Kate, I'd never let him forget that I won him a whole lot of money in that bet!)
Well, they've only started their married lives at the end of the play, but by every indication, Kate. Yes, Petruchio has "tamed" her in ways that seem abusive to the modern eye, but he recognizes her quality, strength, mind, spirit, and beauty. Bianca's husband saw only the beauty. He has to win her, but he doesn't have to put out the effort to do so that Petruchio does. Bianca clearly values him less at the end of the play, acting in ways that cost him money and carping at him. By all period standards, Kate has the best marriage.
I have to concur with my fellow editors on this one. Whilst Bianca spends most of the play appearing to be the meeker and milder sister, she shows her true colours at the end of the play in Petruchio's competition. Having won her man, she now no longer needs to make him feel like the centre of the universe and is free to follow her own selfish desires. Katharina, on the other hand, recognises the rights and responsibilities that come with being married, having been effectively "tamed" by Petruchio. They will definitely have the happier marriage.