Suppose you want to use a famous phrase from a play or novel, such as something from one of Shakespeare's plays. Is it more appropriate to attribute the quote to Shakespeare, or to the character? Does it just depend on the character, and whether or not they're well-known?

It might depend on why you're quoting but I think, in general, it would be better to quote the character, referencing the play with Act and Scene citations.

"[W]hat says Quinapalus? 'Better a witty fool, than a foolish wit.' " Clown, Twelfth Night Act 1 Scene 5.

In this case the character is himself quoting someone, although whom, is an unanswered question.

Sorry, I need to clarify. I really meant in casual conversation, although others may be equally interested in how it applies to writing papers too.

Alex, you're reminding me of the movie Clueless - the scene in which three characters, modern LA teenagers all, are riding along in a car, and one of them says, "It's like Hamlet said, 'To thine own self be true.'" The one riding in the back pipes up and says, "Hamlet didn't say that."

first one: I think I know my Shakespeare. It was Hamlet.
backseat: I think I know my Mel Gibson, and he never said that. It was that Polonius dude.

I'm paraphrasing coz I don't remember the exact dialogue, but that's close enough....I think I'd credit Shakespeare, probably, in conversation - eg:

friend: My family want me to go into law school, but I want to be a doctor.
me: Well - to thine own self be true.
friend: That's good, I like that. Who said it?
me: I did.
friend: But you're quoting someone, right?
me: O, all right. It was Shakespeare.
friend: Play or sonnet?
me: Play - Spamlet.

Or similar. I can't always remember which character said the thing I liked - but I can usually remember if 'twas written by Shakes or some other almost-as-great. And once you know the author, the character whom the author used as mouthpiece is lookable-uppable, for them as are interested.

hey, you know what mg? I agree with you! and the least of reasons is not that I'm more likely to save myself embarrassment through misidentification of a character.

tsuwm agreed with me! My work here is done.

no, seriously though: that's a good point. Although I cannae exactly say why I'd credit the author, rather than the character. Prolly because the credit BELONGS to the author. Duh. It's past my bedtime. Why am I here?!

Duh. It's past my bedtime. Why am I here?!

Who asked that question? Homer Simpson or Matt Groening?

It's an interesting point, though. Who do you credit for a saying? It probably depends in part on (a) if you know who the character was, and (b) if you think that your interlocutor will, too. If you were to shout "Yabbadabbadooooo!" and someone asked you who said that, would you credit Fred Flintstone or whoever it was who wrote the script for Hanna Barbera? But if someone were to ask you who said "parting is such sweet sorrow", would you say Billy the Scrivener; or that it's from "Romeo and Juliet" Act 2, Scene 2; or that Juliet said it?

Which reminds me of a TV spoof on R&J from a long time ago with, I think, Frankie Howerd as a very, very effeminate Romeo. Such lovely legs in those tights! Juliet was someone like Hattie Jacques.

Juliet: "Romeo, Romeo, wherefore art thou, Romeo?"
Romeo: "'Oldin' up the bleedin' balcony, you fat git!"

- Pfranz

Since this thread has wandered beyond the original question and into the Shakespearean realm, I see an opportunity to share one of my favorite bits of humor.
Not many are aware that John Wayne's first attempt on the legitimate stage was a performance in Hamlet. After completely butchering the role, he was boo'd from the stage.
As he faced his audience of critics, he shouted above the boos, "Now wait a minute. I didn't write this shit."

JohnHawaii has reminded me of a related question I have for y'all: If you're talking about a movie with someone, would you talk about the characters and use their names, or would you talk about the actors who portrayed them?

Eg, if you were discussing American Beauty, would you say, "I thought Kevin Spacey was going to shag the teenager," or would you say, "I thought Lester was going to shag the teenager"?

Or

If you were discussing Eveleyn, would you say, "I wasn't sure if Pierce Brosnan was going to get his kids back or not," or would you say, "I wasn't sure if Doyle was going to get his kids back or not"?

Or if you can't remember the name of the character, but you (as I do) at least differentiate between actor and role, would you instead say something like, "I thought Kevin Spacey's character was going to shag the teenager," or "I didn't know if Pierce Brosnan's character was going to get his kids back or not"?

Just depends if I can actually remember the names. I'm more likely to say something like "I thought the father was going to get off with his daughter's friend."

Bingley

This ends up being a personal thang. You could ask Juan about this, but I believe some actors will say, "I did such and such," and some will say, "My character did such and such."