What a fantastic thread! This is one of my most favorite subjects to talk about - and sometimes I can go on endlessly about Mr. Shakespeare, so please excuse the length. Now on to my favorites.
I'll start with the Histories, because they are probably my 10 most favorite of his plays.
King John - the role of the Bastard. Specifically, his reflection on when he gets his new title; the speech starting with "A foot of honour better than I was..." delicious speech and delicious role. Also, how he makes such a huge shift in his character from "outsider and commentator" to active participant. Stunning.
Henry VI part 3 - Specifically Margaret's stuff. I sometimes think that for you to really understand Richard III, you have to see Henry VI/3 first. One of my favorite lines is when York calls Margaret "a tiger's heart wrapped in a woman's hide." Such intense drama!
Henry IV part 1 - This might be obvious, but the scene in the tavern when Falstaff and Hal switch roles. The end of that breaks my heart.
OKAY! Now, to the "Comedies":
I'll admit that I'm not much of a fan of the standard comedy, and though I love Much Ado, Twelfth Night and As You Like It, I think I love the "problem plays" most.
Measure for Measure - one of the most appropriate titles for this play, in my opinion. I don't think that Angelo is widely discussed as being one of Shakespeare's classic "villains" but I really think he is, in a strange and uncomfortable way. When he has his confrontation scene with Isabella (demanding that she sleep with him in order to free his brother) is just harrowing. And the Duke is such a wonderful role too...
All's Well That Ends Well - Helena is a marvelous enigma: strong, strong willed, strong in love, and willing to do anything to be with the man she loves, despite his not loving her. Great role in the Countess as well.
Troilus & Cressida - I've always felt that the love scenes between these two are the more "mature" version of the R&J scenes. When Troilus talks about actually attaining his love and having it reciprocated in Cressida, he says: "I fear it much; and I do fear besides, that I shall lose distinction in my joys." Wow. The broader reason for my love of this play is that it is so wide open with directorial possibilities.
FINALLY! On to the Tragedies.
A combo of Lear and Timon of Athens... they both lose everything (including their minds!...), they both wander into the wild to sort things out in a way, and they both die because of those choices. However, when it comes down to it, how can you not love Timon's speech to the whores (he's giving away all the gold he found buried in the ground) where he admonishes the whores to never give up their profession. He says: "Be whores still. And he whose pious breath seeks to convert you, Be strong in whore, allure him, burn him up; Let your close fire predominate his smoke." Wonderful.
I also love Macbeth... though I'm not entirely sure why...

I like the elements of prophecy and how we can be fooled by the "good news" we hear in it without considering the path that we take to reach it.
Other Honorable Mentions:
Berowne's speech in Love's Labour Lost "A lover will gaze an eagle blind..." Beautiful, quiet, respectful and so moving.
All of The Tempest. Probably not an "honorable mention" then, but it is one of the most nearest-and-dearest-to-my-heart plays. Two speeches in particular: "All the infections that the sun sucks up" of Caliban - just because it's fun to do. And of course, "You do look my son in a moved sort..." with the "our revels now are ended..." and "we are such stuff as dreams are made on." I can never hear (or perform...) this speech without getting a choked tear.
Two Noble Kinsmen - has some beautiful male/male "love" scenes and actually some overall VERY beautiful writing. A strange show, and with many, many levels.
Just a quick mention of a case of "what the--?" In Cymbeline, when Jupiter descends and talks to Posthumus in his dream... (what the?) and then Posthumus's dead family appears... (Kind of like the dream in Richard III, but in a non-curse-and-die sort of way!

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Sorry this post is so long, but I just love Our Will!
I leave you now with one of my favorite insults; from Measure for Measure: “Your bum is the greatest thing about you; so that in the beastliest sense, you are Pompey the Great.”
Thanks again for starting this thread,
Adam