Opie rides his bike on the sidewalk and meets Andy and Barney at the doors to the Sheriff's Office. In Opie and the Spoiled Kid (1963), riding a bicycle on the sidewalk was not allowed and resulted in a reprimand and later, after a repeat offense, in confiscation.
When Barney enters the shed with the dynamite, he has to use his lighter because it is dark. Yet when Jimmy, the goat, was there a few minutes earlier, eating the dynamite, the shed was well-lit. When Andy enters a moment after Barney, it's well-lit again.
Andy reaches down and moves the pencil cup while on the desk. There is no mistake in continuity.
To get free, Jimmy, the goat, pulls the rope from the backrest of the bench, where it was looped around the slats. It drops down but in the next shot, the rope is lying on top of the backrest and is pulled out as he walks away.
When Andy and Barney crouch on the desk, there is a pencil cup full of pencils in front of Andy. But when Andy grabs the rope and steps off the front of the desk, the cup has moved a little to his right.
No one would ever store cases and cases of dynamite in the middle of town and no one would leave explosives unattended without them being securely locked up.
The show obviously takes a lot of liberties with the properties of dynamite. Inside the goat's stomach, the ingredients are all separated from each other, moist, away from any kind of ignition and not compressed inside the tube. The goat would be impossible to explode.
When Andy hands the rifle to the mayor to shoot the goat, Andy says that when the goat falls, it will cause the dynamite to explode. Even if that were true (and it isn't, as noted elsewhere), the goat would explode immediately, upon the impact of the bullet.
Andy comes out of the shed with Jimmy's rope, but the way it had been tied around his neck would have prevented it from coming off because his long horns would have prevented it from slipping around them.
At one point Andy puts the rope around the goat's neck but just lays it there and doesn't tie it.
At the end of the scene when Andy and Barney encounter Opie riding his bike and Opie tells them where he saw Jimmy heading, Andy and Barney try to walk past Opie. Andy Griffith and Don Knotts get stuck trying to go around the bike and Ron Howard slips off the bike before the scene cuts.
When Barney first shoos Jimmy, the goat, out of the courthouse, Jimmy turns to the right from the courthouse, in the direction of Floyd's Barber Shop. Not more than a moment later, Andy enters the courthouse from the direction of Floyd's Barber Shop (where he had been in the previous scene). Yet, he apparently never saw Jimmy and said so when Cy Hudgens came in looking for Jimmy.
The mayor is accused of wanting to bring the bypass past his brother's filling station, yet Wally, no relation to the mayor, has the only filling station in town.
The story takes place long before speed dialing was invented. Miss Vickers' call to the courthouse comes immediately after another construction blast, with no time at all to pick up the phone and say, "Sarah, get me the courthouse."
The construction crew's blasting engineer claims the goat could possibly explode if he falls down, after having eaten dynamite. Dynamite needs something to set it off, like blasting caps with a lit fuse or an electrical charge.