Revealing mistake: During "Brand New Day", watch the background as Dr. Horrible appears as a giant and stomps down the city-street. As he takes his first step, a piece of concrete from building to his right falls off and hits the ground. However, if you look closely, it's obviously a quickly-made computer effect. As the piece of concrete hits the ground, it simply vanishes, and is replaced by a rather odd-looking animated "poof" of dust. The issue is the concrete piece was much bigger than the cloud of dust, so it's clear that the piece simply disappears right as the dust-poof appears.
Continuity mistake: During "Brand New Day", at one point, Dr. Horrible stabs a picture of Captain Hammer attached to a dartboard with a handful of darts, all right in the center of the picture. A few shots later as Dr. Horrible sits, suddenly the darts are far-more spread out. (It's in the background and can be hard to see, but it's noticeable.)