What examples of the Socratic Method can be found in Socrates' Apology?
The so-called "Socratic method" is a means of philosophical enquiry, wherein people are interrogated about what they have said and subsequently worked through several related questions to see if they contradict themselves. If they do, this proves to the person that their original position had been inconsistent, and in the end, they are wiser than they were before, because they have been made aware of their own ignorance. Socrates calls this method elenchus, and it gave birth to dialectic, the idea that one cannot arrive at the truth without first interrogating one's own position and modifying it accordingly.
In Apology, Socrates lays out this method to the Athenians using a number of examples. He describes how a friend had approached the Oracle at Delphi to ask if any man were wiser than Socrates, and he had been told that no man was wiser. Socrates tells how he then began to seek out people reputed to be very wise, and determined that their reputations were unjustified by forcing them to confront their own inconsistencies. Socrates's reasoning is that he himself is indeed the wisest man in the city, because, unlike every other so-called wise man, he recognizes his own limitations and ignorance.
When Socrates is found guilty, he goes on to use the Socratic method to question the wisdom of his being put to death. After all, he argues, nobody knows whether death is a blessing or a curse; consequently it would be inconsistent of the jury to put him to death with the intention of punishing him.
What is popularly termed the "Socratic method" and more properly termed elenchus is a method of discovering the truth by dialogue in which propositions are put forth, contested, and either shown to be legitimate or, more commonly, disproven, leading to a state of "aporia," or uncertainty. The Apology, being a courtroom speech, actually uses elenchus far less than most Platonic dialogues; it consists mainly of a long lecture with one short dialogic section; otherwise it describes Socrates's habits and methods but does not actually exemplify them.
The one actual example of elenchus in the speech is the dialogue with Meletus in which Socrates attempts to clarify the basis of the accusations against him. Socrates refutes two of Meletus's main points. First, he questions Meletus to determine that corrupting the youth of the city would harm Socrates himself, and this argues that the accusation that Socrates corrupts the youth intentionally must be wrong. Next, he demonstrates that Meletus contradicts himself when he says:
First that I don't believe in gods, and then again that I do believe in gods; that is, if I believe in demigods. For the demigods are the illegitimate sons of gods . . .
Socrates uses his trial not to prove his own innocence, but to expose philosophical truths to the citizens of Athens, as well as bringing to light the true motives of his accusers. While the Apology is more of a speech than a dialogue, there are multiple examples of the Socratic method. First, he describes how he went about understanding a declaration of the oracle at Delphi, which was that he was wiser than all men. Socrates tells how he questioned various Athenians, from the wealthiest and most powerful to the humble craftsman. From this he arrived at the truth, which was that he was wiser than others because he was willing to admit what he did not understand. In addition, the philosopher was afforded the right to examine his accusers, and when he questions Meletus, one of the three men who brought him to trial, he uses his method of incisive questioning to show that the allegations were rooted in his accusers' desire for revenge against Socrates rather than the conviction that he was actually guilty of a crime.