Here are four very popular ones. The one below shows that for the first time Jack has been honest about something without even trying nor wanting to. Since his "christian name" would be Earnest, like his father's, he had been technically telling the truth all along, which even was a shock to him.

Jack: On the contrary, Aunt Augusta, I've now realised for the first time in my life the vital Importance of Being Earnest. (III.180-181)

(Act 3)- This is the part when Earnest discovers his father's name was Earnest, making him a true Earnest after all. The way he uses this pun is sort of to justify himself to Gwendoly and her mother, and perhaps to give himself a squeeze for actually telling the truth for once.

I always told you, Gwendolen, my name was Ernest, didn't I? Well, it is Ernest after all. I mean it naturally is Ernest. (III.170)

Algernon figures that marriage is a demoralizing state, and throws a pun at it while commenting with Lane and placing emphasis on the champagne in married homes.

Algernon: Why is it that at a bachelor's establishment the ... servants invariably drink the champagne? I ask merely for information.

Lane: I attribute it to the superior quality of the wine, sir. I have often observed that in married households the champagne is rarely of a first-rate brand.

The origins of Jack back in the railway station make Lady Bracknell mock him, and as far as his possible relationship with Gwendolyn, she remarks to the end of their acquaintance as a "Terminus", a double entendre which means both "end" and "terminal station" as in the one where he was found.

Lady Bracknell: Mr. Worthing, is Miss Cardew at all connected ... with any of the larger railway stations in London? I merely desire information. Until yesterday I had no idea that there were any families or persons whose origin was a Terminus. [Jack looks perfectly furious, but restrains himself.] (III.61)

Hope this helps!