When Penny tries to tell maybe there's more to Captain Hammer than meets the eye, Billy brings up that maybe there really isn't and he really is like the first impression he makes deep down too. "Like with pie."
Dr Horrible managing to give a credible look of bemusement at Captain Hammer's "gently wafting curtains" line - whilst being strangled, along with an appropriate sound that would have come out as a Flat "What.".
Dr Horrible bemoaning the fact that Operation Heist didn't go as planned at the end of "A Man's Gotta Do": BALLS! Smash to Black.
"Did you notice that he threw you in the garbage?"
"What heist were you watching?"
"The Evil League of Evil is watching so beware/The grade that you receive will be the last we swear!/So make the Bad Horse gleeful, or he'll make you his mare/You're saddled up/There's no recourse/ Hi-ho Silver!/Signed, Bad Horse."
"He saw the operation you tried to pull today/, but your humiliation means he still votes neigh/, and now assassination is just the only way/, there WILL be blood, it might be YOURS, so GO KILL SOMEONE!/ Signed, Bad Horse." Keep in mind the second was a phonecall.
How can you mention Dr. Horrible here without this?
Captain Hammer: She's giving it up. She's giving it up hard. Because she's with Captain Hammer. And these... (holds up his fists) ...are not the hammer.
(He leaves the frame for a moment while Billy stares after him.)
Watch Neil's facial expression during that scene: he's biting his lip, clearly trying to keep from laughing.
Billy gets one just before this, when he learns Captain Hammer is going to stop by the laundromat:
Billy: (tries to look at his watch, but isn't wearing one) Oh goodness, look at my wrist, I gotta go.
Penny: But what about your clothes?
Billy: (turns back, glances in washer) I don't love these. See ya!
Even better is that the washer Billy opens is a different one than the one he and Penny were sitting on before, which implies he didn't even check the right washer.
Half the exchanges between Penny and Billy at the Laundromat:
Billy: I want to be an achiever. Like Bad Horse.
Penny: The Thoroughbred of Sin?
Billy: ...I meant Gandhi.
Penny: You're driving a spork into your leg.
Billy: So I am. Hilarious.
For me, it has to be the part where Dr. Horrible pauses in the middle of his Moment of AwesomeVillain Song to correct a cowering reporter on the spelling of his name. After singing about how there will be anarchy... that he will rule.
Also, pretty much all of the 'commentary'... the fact they wrote a musical (pseudo) commentary possibly counts as a Funny Moment... but especially the 'sea shanty' exchange.
NPH: Songs can hurt like a fist.
And immediately after the opening number:
NPH: A muscial blog is a zany idea, that's wacky! Joss, where did it come from?
Joss Whedon: It came from pain.
NPH: Uh, let's not talk to Joss. He's scary, and confusing.
The Tur-Mohel's application to the Evil League of Evil. The application requirements include a minion... so Tur-Mohel has a Minyan (a group of ten adult Jewish males). And that's just the tip of the iceberg.
The revelation of Bad Horse. Admit it, how many expected he'd actually be a horse?
"Oh, I'm in pain! I think this is what pain feels like! Oh, mama! Someone maternal!"
Penny: After years of stormy sailing/ have I finally found the bay?
Billy declares his application is strong, in part due a letter of condemnation from the deputy mayor. This opens so many questions. Did Billy ask the deputy mayor to write a letter describing how much s/he doesn't appreciate all Dr Horrible does? Is there a letter starting, 'I'm being forced against my will to write this letter, although, despite the force, I do genuinely condemn Dr Horrible.'? Did s/he pen an article decrying Dr Horrible's antics, and Billy decided to send a newspaper clipping in? Whatever the answer, it's hilarious to ponder these things.
Also, because the letter was from the deputy mayor, that implies that Dr. Horrible wasn't horrible enough to warrant a letter of condemnation from the mayor him/herself.
Even the actual commentary (not the musical one), gets in on the act. When Nathan Fillion is honestly and sincerely thanking Joss Whedon for the opportunity to appear in this project, Joss replies "No one bring the smarm like you. And your acting's good, too."
Billy's reaction to Penny appearing behind him with her petition.