When Situational Irony occurs in theater the audience is expecting the play to progress in one direction, when in fact the story or the situation takes an unexpected turn. Situational Irony deals with usually sharp contrasts from expected plot lines and several contradictions. Shakespeare also uses Verbal Irony and Dramatic Irony in his plays.
To understand the Situational Irony in the first scene of Taming of the Shrew you must be able to differentiate Verbal and Dramatic. Verbal Irony is exactly what it sounds like: It is using a phrase that means something different on the surface than your intended meaning. Modern day, we refer to this as "Sarcasm." "I just love eating liver and Lima beans on my birthday."
Dramatic Irony is where the audience has more information than the character. The best example of this is in Romeo and Juliet. The audience is fully aware that Juliet is only asleep when Romeo enters the crypt and commits suicide. They, powerless to stop the situation from happening, are in the clutches of Dramatic Irony.
Situational Irony, which is what is being observed in Act 1: Scene 1 of Taming of the Shrew, must have two aspects- First the audience must make an assumption and be certain they know where the plot is going. Second, the assumption must be not only incorrect, but it must clash with the plot in a way that the audience becomes interested in where the new plot will go.
An example of this in Act 1: Scene 1 involves Lucentio and Tranio. First, the beginning of the play indicates that Lucentio is a young man who is supposed to be in Verona to go to school, but is led into partying by his servant. Even the introduction of Bianca gives a preconceived notion that this play is going to have a Romeo-esque young man chases young girl plot. The roles of the young, love stricken noble and his loyal (somewhat comic) servant are well defined and the audience is ready to enjoy the comedy.
It is at this point, Shakespeare uses Situational Irony to suddenly change the direction of the play when Lucentio learns he cannot go near Bianca because of her shrewish sister. The traditional roles of the characters are switched- Lucentio is going to pose as a tutor for Bianca and woo her under her father's nose. His man servant Tranio will be posing as Lucentio. Tranio pushes the Situational Irony by pointing out to the audience that Lucentio has completely ignored the huge obstacle in the way (Kate). Tranio's mock reluctance to live the life of a noble for a period of time, once nudged into acceptance, silences his warnings that this might not be as easy as Lucentio thinks it will be.
This Situational Irony in the scene provides two intriguing questions for the audience and draws them into the plot:
1) Will Lucentio be able to present himself as a lowly tutor? (Keeping in mind he's been sent to Verona to study.)
2) Will a lowly servant be able to convince others that he is in fact a noble?
To add to this juicy bit of plot, enter Biondello. A somewhat dim boy who is being pulled into the switch. This presents a third question?
3) How is Biondello going to ruin everything? (This in itself is situational irony, because Biondello is not the foil for the charade.)
A second bit of Situational Irony in Act 1: Scene 1 is really a setup of Situational Irony that will be fulfilled in the next two scenes-
The two suitors of Bianca: Gremio and Tranio have been forced into a position where they must get Kate married before they can fight for the hand of Bianca. They enter into an agreement to find a man who would be willing to marry the shrewish Kate for money. They both agree that to find a man that would be willing to do this will be incredibly hard to find, but they agree to work together to make this happen. In the mean time they agree to call a truce. This sets up the Situational Irony that once Petruchio is found to marry Kate, Gremio tries to subvert this deal by being a tutor for Bianca. Petruchio foils this by instead giving Gremio as a tutor to Kate, making Gremio's life miserable and his head hurt from being beaten by various musical instruments.