Where are the puns and epigrams in the play?Who speaks the puns?
The puns mostly surround all the conversation regarding the name Ernest and the adjective earnest. Most are said by Algernon, but at least each one of the characters has a go at a pun or an epigram.
The epigrams are the contradictory phrases that most all the characters say trivially such as Lady Bracknell saying that innocence should be preserved as a delicate fruit, for if you "touch it" (get educated) it would lose all its appeal.
Jack, in the end, after basically accepting that he had lived a double life and that his real name was not Ernest, quickly checked his father's records only to find out that his father's name was Ernest and, for that reason, he was "dully" named Ernest. After realizing this, he said that for the first time he learned the importance of being earnest. This means that he is again denying that he ever lied since, after all, he had been "Ernest" all along, hence making him an "earnest" person despite of what he did before.
Algernon says many puns especially regarding his family, saying that only family members and bill collectors would knock on one's door with such annoyance. He also puns to the name of Ernest telling Jack that he is "the most earnest looking person he's ever seen in his life."
Another big pun regards women, when Jack claims that Cecily and Gwendolyn will call each other "sister" and Algernon replies that such thing will only happen "after they call each other many other things."
In all, everything regarding the name of Ernest is the main pun of the story, so whatever comment is made on earnestness is a direct pun to the triviality of the play.