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(During "My Eyes"/"On the Rise") At the soup kitchen, Dr. Horrible (wearing a fake moustache) replaces the soup serving guy. He misses the bowl, ladling soup right back into the pot, while spying on the date.
The cell phone rings with the "Bad Horse Chorus" theme.
(In "So They Say") "Next up: Who's gay?" The fanboy sings the next line (with a lisp).
(In "So They Say") Penny waits in vain for Billy at the laundromat, with two frozen yogurts that she brought. She turns to look when someone comes in, and seems disappointed that it's not him.
The dry cleaning bill is laminated.
(In "Slipping") The line, "Anarchy, that I run!" (Hah!)
Dr. Horrible couldn't bring himself to kill Captain Hammer. He spent a whole song psyching himself up to do it, but kept hesitating. "There's no time for mercy." [Tries to get himself to pull the trigger.] "Here goes no mercy! [Dejectedly gives up.] Contrast this with Capt. Hammer casually pulling the trigger shortly afterward.
Captain Hammer gives Dr. Horrible the finger when he sings, "WAY!" at the end. It was the last word of the song that he hadn't finished.
During the bank robbery, the money bags keep slipping out of Moist's hands.
Penny is the only good person among the three. Both Dr. Horrible and Captain Hammer cite her as a good influence, but they see that influence only through the lens of their own warped paradigms, and do not actually change themselves. ("Head up, Billy buddy: There's no time for mercy.") ("She taught me... It's not enough to bash in heads: You've got to bash in minds.")
In an obvious way, this is a "nerd vs. jock" story. Neither is inherently good: The "protagonist" is whichever the point-of-view character happens to be. That's the nerds in "Revenge of the Nerds," and it's the jock when Superman confronts Lex Luthor. Both ideals are examples of toxic masculinity. Where the jock wants to be the warrior king, the nerd wants to be the philosopher king.* This story has both competing for the same two prizes: Power in their city, and the affection of a pretty and powerless woman.
* I've borrowed that phrasing, and if someone can figure out where I read it, I will gladly credit the author.
Penny's powerlessness is highlighted not only by her lack of agency in the competition for her, but also by her own quest: She spent a great deal of time and effort helping the homeless and gathering signatures to convince the mayor to donate an abandoned building to the shelter. The latter task was trivial for Captain Hammer, who put in no effort at all but had the ear of the mayor.
This story is also a "nice guy" fantasy. In Internet parlance, the "nice guy" is a shy person who doesn't know how to talk to women, and who believes that he is single because women prefer assholes. He wrongly believes that he respects women. He may befriend a woman in hopes that she will fall for him, while never indicating his own interest... Until, perhaps, he gets frustrated with the lack of results, insists on a relationship, and skulks away from the friendship while making disparaging comments about her after she turns him down. Dr. Horrible's story plays out like a story written by a "nice guy" about a "nice guy". Through the power of his shy charm, Billy gradually befriends Penny. Despite her visceral attraction to Captain Hammer, she begins to fall for Billy. They bond and nearly kiss over frozen yogurt. Captain Hammer reveals himself as nothing more than an asshole who objectifies and doesn't really appreciate Penny: First to Billy at the laundromat, and later inadvertently to Penny, who quietly leaves him at the conference. For just a moment, before everyone gets "Jossed"*, the fantasy of the "nice guy" is complete.
* A surprise loss/death, considered occasionally characteristic of Joss Whedon's productions (though two examples were actually forced by Fox network execs).
This view of the "nice guy," while often depressingly accurate, is necessarily incomplete. It neglects the fears, inexperience, and motivations (some of which are grounded in his idea of respect) that hold the "nice guy" back from respectfully pursuing romantic relationships. Those factors also affect genuinely respectful and nice guys who may be mistaken for "nice guys": A confusion which exacerbates the problems common to both. I wrote about this back in 2013. I would have made some different choices in my writing later, to be more conscious of what people might mistakenly take from subtext before they get where it's going, but I hope it still works well enough.
Doctor Horrible has two quests competing for his attention: Gain power over the populace, and gain Penny's adoration. He wants both, but when he is forced to choose, he leaves Penny behind in favor of the wonderflonium (at both the start and end of the song). This presages the end of the show: He gains what he pursued, and loses what he did not pursue.
This brings us to the trope called the "Woman in the Refrigerator". It appears especially in comic books, and "Dr. Horrible" is a comic book in live action miniseries form. In this trope, the writer kills off a woman in order to give motivation or a narrative to a man. This treats the woman as disposable: Her entire life is worth less than the man's story. Penny dies so that Billy can be sad. If you don't know why this is bad, watch the video in the link; it's only 6.5 minutes.
If you don't know why misogyny is bad, you can count Joss Whedon as company, and it's a damn shame on both counts.