Arthur's horse, including tack, would change dramatically from scene to scene even when on a multi day quest where there would be no opportunity to switch horses.
The time it takes to travel fixed distances often varies during episodes. Morgana (and others) sometimes ride between two places in a few hours while others need to stop to make camp for the night at least one time while traveling between the same places.
There's an episode where a knight of the round table can be spotted with a plastic water bottle.
Constantly throughout the show we see mail clad knights being injured or killed by slashes from a sword. It's nearly impossible for a sword to penetrate chain mail and cause anything more than a bruise or broken bone, especially when slashed with the swords they're using. Even a very hard thrust with a sword against mail would have minimal effect. They might be able to break a few links with a very hard slash, but would never get through it all the way to the body.
Arthur's sword is always shown either inside a leather scabbard or tucked into his leather belt. Yet throughout the entire series, each time he grabs his sword to wield it, it makes a distinctive metallic scraping sound, as if withdrawn from a metal scabbard. This is physically impossible.
Throughout the show Giaus wears glasses (in S1E1 he even mentions he can't read without them) but they weren't invented until 1290. That is at least 600 years after the possible rein of Arthur.
Throughout the program, Merlin and Gaius seem to have detailed knowledge on things such as infections and the circulatory system, despite these discoveries not being made until over 1000 years later.
On many episodes there are tomatoes present during meals. Tomatoes, being native to Central and South America, did not arrive in Europe until sometime after Columbus discovered America.
There are some episodes where Merlin can sense the use of magic and some where he can't. It is never explained why he can only sense some magic sometimes. However, it would change many episode plots if he could always sense it.
In Le Morte d'Arthur, Merlin gets the cup of life to save Arthur, then Gaius. The cup can save anyone yet as the scene ends, the cup is clearly sitting on the alter. Why Merlin doesn't take it is never explained and would have prevented Arthur's death in the series finale.