In “HMS Surprise,” IIRC, Aubrey is talking to one of his crew about pirates. The man had earlier sailed with another Royal Navy captain who once came across a small ship with a suspicious-looking crew off the coast of Africa. The Royal Navy crew was in favor of hanging them all immediately as pirates, “…but the captain, being a magistrate when ashore, had some scruples about evidence.” As a magistrate myself, I love that line. 
Diana: “Oh. I thought it was the horse.”
I’m paraphrasing, but it’s where Stephen is explaining the difference between a moral philosopher (Professor Graham) and a natural philosopher (himself). Jack carefully crafts his pun before he launches it:
"So Stephen, would it be safe to say you are an immoral philosopher?
“What stuff. Next I will hear you describe Professor Graham as an unnatural philosopher.”
And to move away from the humorous lines, I love the bit in The Commodore when Stephen comes back home after leaving the apparently autistic Brigit in Padeen’s care, and she greets him:
“God and Mary be with you my Father.”
“God and Mary and Patrick be with you, my Daughter. We are going to Spain, the joy and delight.”
I love that part. I’m training my daughters to greet me that way.
“He who would pun would pick a pocket” is one of the few I’ve managed to use in casual conversation.
YES! I about laughed myself sick at that line.
Another favorite of mine is the scene in The Surgeon’s Mate when Stephen makes his speech before L’Institut. Unaccustomed to speaking in front of a crowd, he fumbled, mumbles, and stutters through his presentation. A member of the audience, sent to observe with a mind to determine the truth about Stephen’s role as a spy, comments “If that man has anything to do with intelligence…then I am the pope!”
Others in the same scene are pondering what, if anything, a devastating creature like Diana Villiers can possibly have to do with such a milksop as Dr. Maturin. Someone comments that he is merely her physician, and that he examines her in a completely professional and discrete manner. To which some wag replies, “he must be a paederast.”
I don’t recall which book it is, but Stephen greets a fellow intelligence agent who has also been his patient. And though he has come from the other side of the globe and no doubt has business of earthshattering importance, his first words to the gentleman are: “Ah, Mr. X. And how is your penis?”
Funny how I’m not normally one for scatological humor, but most of my favorite O’Brian scenes are bathroom/bedroom jokes.
Context, please…?
How about this scene with Jack & Stephen, from The Truelove:
“Pray take off your breeches and bend over that locker,” said Stephen, sending a jet from his enema through the open stern window. A little later, and from this position of great moral advantage, he went on…
(to lecture Jack about the mood of the crew regarding Jack’s intent to turn ashore a midshipman and his girl, ending the lecture with: )
…You may put your breeches on again."
“Damn you, Stephen Maturin.”
“And damn you, Jack Aubrey.”
Post Captain.
Mr. Babbington is taking Diana to Jack’s party in some form of horse drawn conveyance. He is perhaps too forward, and she needs to “take him down a notch.” The “bean fed horse” farts, Babbington says “Pardon me.” and diana replies, “I thought it was the horse.”
I see. Thank you. Didn’t Seinfeld have some fun with a farting horse once - IIRC it was Kramer’s, when he was giving tourists carriage rides around Central Park?
I am proud to say that I have never seen Seinfeld.
Where, oh where is the Jewish guy smiley when it is really needed?
For those of you who are struggling with the books, I heartily recommend the audiobooks, particularly those narrated by Patrick Tull. They’re marvelous to hear!
My favorite line (and, remember, I heard this, so it sounds funnier to me than just reading it), came when Jack risks arrest for debt by going ashore for Mr. Babbington’s celebration party. He’s spotted, and the bailiff and his men try to enter the room.
A grand meelee follows. Jack escapes out the window and runs down to the shore, hallooing for the sailors. They come to his aid, and before they clash, the bailiff orders Jack to “halt in the name of the law!”
“FUCK THE LAW” the sailor screams.
As has been said, there’s very few curse words in the entire series, and its sudden appearance (luciously expressed in that Brit accent), made me laugh so hard I had to stop the tape until I could compose meself.
Seriously, check out the audiobooks. I listened to the entire series through my library, and while I have some of the books, I rarely read them.
I believe Jack then has his sailors impress the baliff and his thugs. 
Not a specific line, but one of my favorite scenes was in Post Captain, when the reader is presented with a charming scene of a bedraggled traveling gypsy whose chained bear is urged by a group of bored onlookers to perform a few choice dance steps. I read the scene to my 8 year-old daughter, who was quite astonished (as I had been) when the bear, once out of hearing range of its unwanted audience, speaks a few words of complaint to its handler.
Members of the inner brotherhood will have no difficulty recalling the incident.
“Aubrey, pass the salt.”
Also I’ll note, the profanity was few and far between. When Stephen was stranded on an island while the ship was beating to and fro to get back to him he was baking on a small rock island watching birds. Jack asked what he drank, Stephen answered Boiled Shit. The shock from Jack gave me pause. He had never heard Stephen give a blunt profane reply.
“May I trouble you for the salt, Sir?”
“The bottle stands by you, sir!” – I actually got to use this at the dinner table a while back. My brother (who is a fan of O’Brian as well) was over, and since I now drink wine, we had a bottle open… and it was by him, and I wanted it.
I think this is from Post Captain. An ape has been delivered to Melbury in Stephen’s absence. Jack informs Stephen that there is an ape waiting for him:
“An ape? What kind of ape?”
“A damned ill-considered ape. It has been offering itself to Babbington.”
It was a sad day for readers when Babbington was commissioned and left Jack’s service.
Stephen, to a guest at the dinner table:
“…Tell me sir, do you know of any banker that really understands his business? Some modern Fugger?”
“Oh, Stephen, if you please,” cried Jack, for both Edward and Henry Smith, the sons of an Evangelical parson they much admired, were spoken of as Blue Lights in the Navy (prayers every day aboard and twice on Sunday)… and he did not wish his guest to be offended.
“I refer to the Fuggers, Mr. Aubrey,” said Stephen, looking coldly at him. "The Fuggers, I repeat, an eminent High Dutch family of bankers, the very type of those who understand their business, particularly in the time of Charles V.
“Oh? I was not aware - perhaps I mistook your pronunciation. I beg pardon.”
Perhaps it wasn’t the same conversation, but I seem to recall Jack saying, “My bankers are Hoares.”
Would that be a rather oblique way to say, “Please pass the wine bottle,” then?