These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Also, whether it's a multiple personality or not, is Billy a genuine shy nice guy, or a passive-aggressive Nice Guy (TM) who, while less openly selfish in his pursuits of relationships than Captain Hammer, is not really any more interested in her as a person? It's a good story either way, but a very different story.
Penny, on one hand, she dates a guy who's not very subtle about what a giant Jerk Ass he is because he's famous and can help her with her personal goals, she's possibly emotionally cheating on said Jerk Ass with Billy, and she quite possibly uses her last words just to twist the knife in someone's gut. The view of her as being full of Incorruptible Pure Pureness is more Billy's projection, and looking at what she actually does shows her as flawed and shallow like everyone else in the story.
Then on the other hand, should she really be able to see Hammer's jerkiness? He's fairly low-key around her, until the press conference, where she sees it and doesn't like it a bit. Is she just using Hammer to achieve her own personal goal? Since her goal is a to help the homeless, is the manipulation unjustified?
Ear Worm: The songs are — dare we say — horribly catchy.
Ensemble Darkhorse: A literal case here with the Thoroughbred of Sin himself, Bad Horse; despite only featuring in two short songs and having a brief physical cameo at the end, just about everyone loves Bad Horse.
Faux Symbolism: Dr. Horrible's overlarge armchair in which he plots world domination. Either he feels inadequate or it's just for comedic effect.
See the Jossed entry in Trivia—that chair just happened to be in the house where they were filming.
Fridge Logic: When Billy spies on Penny, at one point he sees her eating frozen yogurt.
Foe Yay: Between Captain Hammer and Dr. Horrible, who seem to take their relationship very seriously:
Dr. Horrible mentions at the very beginning of his blog that Captain Hammer is his nemesis, not some poser in a parka. Captain Hammer apparently returns the feeling as he admits in his song that everyone's villains aren't as cool as his.
"These... Are not the hammer. The hammer is my penis." One act later, Dr. Horrible taunts him with "Hammer, meet nail!" If that's not a Metaphorgotten, it's a very revealing Freudian Slip.
Harsher in Hindsight: The fan reaction to Loki in Whedon's film of The Avengers is eerily similar to how the public react to Dr Horrible in the ending, especially considering that they had the same motives (taking over the world in an effort to make it a better place) - Loki kills hundreds of people, but he's still worshipped as a villain, just like Dr Horrible was, complete with obsessed fangirls.
Misaimed Fandom: Some people, as highlighted in this essay, are arguing that a large part of the audience seems to have missed the point especially in the way it tackled the idea of Women in Refrigerators and of Billy as a Nice Guy. Billy's complete transformation into the Dr. Horrible persona is clearly and unambiguously portrayed as a tragedy; not only because of the loss of Penny, but because it costs him the very humanity that his friend associates with. Some people are still going to prefer the Well-Intentioned Extremist Dr. Horrible over the nebbish Stalker with a Crush Billy.
More worryingly, some Youtube comments imply that a few people wish they could deal with their bullies the way that Billy goes on to do. This is slightly worrying, to say the least.
Moral Event Horizon: Captain Hammer dating and sleeping with Penny just because he knows it'll hurt Billy/Dr. Horrible. He might be a superhero, but this moment makes it clear he's not in it because he's a good person. This is ultimately what causes Billy to plot to kill him- not the countless times Hammer's foiled his crimes and beaten him up.
Tear Jerker: (Dr. Horrible) [singing in Act III] "And I am fine." (The look on his face and the camera's pull-away show he's anything but. He's hurting and alone.) and [singing at the end of Act III] "And I won't feel..." [the door slams shut] - (Billy) "...a thing."
The Woobie: Not only does Dr. Horrible get roughed up on a regular basis but takes it in stride ("...honestly, I'll live."), he suffers the emotional equivalent of a punch to the gut several times. "Penny's Song" establishes Penny as this to some extent, although she's far too optimistic to let it stick.
Iron Woobie: He's taken a ridiculous amount of abuse, and he doesn't remotely look like it. Hammer threw a car at his friggin' head!