Continuity mistake: The Doctor can be seen wearing glasses in some long shots of him riding a motor bike. Normally, Sylvester McCoy (who played the Doctor at this time) removed his glasses before filming. Seems as though he either forgot, or could not ride a motorbike without them.
Continuity mistake: Due to some sloppy editing of the "cliffhanger ending" at the end of episode 2, and again at the start of episode 3, the Doctor, Mel and Burton are seen to freeze in place for no apparent reason, just BEFORE we see the Bannermen line up and aim their guns at them.
Plot hole: At the end of episode 1, the 'cliffhanger' ending is that the Doctor climbs over some railings, and then, by looping his umbrella on the railing, he begins to climb down. He looks down and it seems to be a very long way down. This is resolved in episode 2 by having Glitz walk around and let the Doctor step on his shoulders. Not only is it silly, but because Glitz walked around, it also makes the previous 'cliffhanger ending' totally meaningless.