When Dr. Horrible connects into the electronic piece he threw onto the van carrying the Wonderflonium, his hand-held device reads it is 5:00 PM. A few minutes later, after changing into his outfit after the start of "A Man's Gotta Do," the same device reads 4:57 PM.
After the song "Bad Horse Chorus" advising him to commit a crime to impress Bad Horse, Dr. Horrible folds up the letter. In the next two shots, he's holding the letter unfolded.
When Captain Hammer is choking Dr. Horrible, Dr. Horrible's right hand positions change during shots: first his hand is on top of Hammer's hand, in the next shot his hand is away from Hammer's, then it is back on Hammer's again during the next shot.
Moist retrieves Dr. Horrible's mail, but the letter from Bad Horse is totally dry when Dr. Horrible opens and reads it.
Right before Billy is going to take control of the van, he ducks down in a stairwell and re-emerges as Dr. Horrible. This was either supposed to be instantaneous or only after a minute or so; meanwhile, it has darkened a little, meaning much more time has passed. If something as important as Wonderflonium requires a guard to transport it the guard would not take that much time to leave the area in order to reduce the possibility of a theft.
When Billy is addressing the viewer on his blog, it is evident that his computer monitor is not actually on, since his face becomes darker as it gets closer to the monitor. Had it been on (as it would be were he actually making a video blog), his face would have gotten brighter as it neared the monitor.
When Captain Hammer is riding on the van and the shot is of him from the knees up, you can see a safety wire to the left side of the frame.