Factual error: When Stottlemeyer and Randy are at the painter's daughter's home they find a picture showing him in front of Carson papers, which leads the Captain to say: they have a contract with the treasury, they are producing the paper for the U.S. currency. This is wrong: the treasury is not in charge of printing the Federal Reserve Notes. Rather it is the privately owned Fed that prints the Federal Reserve Notes and then lends it to the treasury at interest. And the interest added is the reason why there will never be enough money to pay the interest, i.e. the state debt.
Character mistake: While the concept of the knife height and stab wound is OK, the way Monk measures the height of the knife is way off. He says it's 48", which is 4 record lengths, but he doesn't put the bottom of the record where the top previously was, sometimes he only moves it half way up. Then the top of the record is above the knife, so it couldn't be 48" even if it was done right. As particular and accurate as Monk is, he wouldn't make this mistake.