Continuity mistake: Farmer Hoggett is an old fashioned man, and naturally his fences are old fashioned and wooden. However, in the scene where the two sheepdogs are rounding up the sheep, the camera follows one of the dogs for a shot, and you can see the fence in the background is much more modern - it's called New Zealand fencing, I think - where in the shot before it was wood.
Audio problem: There are a few points throughout the film when the animals talk, but their mouths don't move.
Continuity mistake: When duchess is thrown outside, in the close-up shot you can see that she almost had the door handle, but the next shot shows that it's higher than her paws.
Continuity mistake: In the scene where farmer Hogget is filling out the application form for the sheepdog trials. There is a postal address line on the form. In the next shot it is gone.
Other mistake: The competition clock at the sheep dog trials has no minute hand.