Several examples within the film (pretty much any time Neil Patrick Harris opens his mouth to sing, for example), but really, the very existence of this is a Moment of Awesome for all involved. Joss Whedon and company cooked up the whole thing to circumvent the issues brought up during the WGA strike; it was funded with Whedon's own money and most of the cast and crew worked for nothing (other than the possibility of the film earning enough to actually pay them). Then the strike ended, and they went through with it anyway, and it actually worked—so much so that when the first act went live online, the influx of viewers crashed the server.
Not to mention that, despite being available for FREE online, the project made enough DVD and iTunes sales that the cast and crew were not only fully compensated for their work, but they apparently also made a profit off of it. Awesome indeed.
It is also worth mentioning that the first act features Captain Hammer, played by Nathan Fillion, falling from the sky onto a moving vehicle and bursting into song.
"Now Doctor Horrible is HERE. To make you quake with fear. To make the whole world kneel." Coupled with Dr. Horrible's Evil Costume Switch to the red lab coat, black gloves and goggles pulled over his eyes.
Third...ed, I think... anyway, you can almost feel the amount of menace dripping off the first part of the song and it. Is .Awesome.
For Doctor Horrible himself, it has to be "Everything You Ever." Summed up by the final lines:
"Now the nightmare's real/Doctor Horrible is here/To make you quake with fear/To make the whole world kneel...."
The very existence of this is a Moment of Awesome. Joss Whedon produced a stellar, tear jerking tragicomedy you can write English dissertations on, with excellent music you'll never get out of your head, all on a shoestring budget— because he was bored.