While Jamie and Murtagh are practising with swords and knives, one of the brief overhead shots shows both of them with their weapons in the wrong hands. The shot has been reversed.
When Claire first appears in the red dress she has a small red fan. She then leaves for Versailles. When she is next seen she has a large printed fan.
When Jamie arrives home from sword practice his hair is long enough to hang past his eyes and reaches his collar. All of a sudden in the last shot - boing! His hair is shorter and very curly.
During a narration, Claire refers to Louise de la Tour as "Marquis de Rohan," pronouncing it the French way: "mar-kee." This is wrong. "Marquis" is the masculine; Louise's correct title in French is "Marquise," pronounced "mar-keez." Nor is the mistake explained by Claire using the English pronunciation of marquis ("mar-kwis"); and even if she was, the English version of the feminine marquise is "marchioness."
The Duke of Sandringham says 'serendipitous'. The word 'serendipty', from which it derives, was only coined ten years later in 1754 by Horace Walpole.
When Jamie and Claire are speaking to the French Minister of Finance, Monsieur Duverney after he has just finished hitting on Claire and embarrassing himself, he says to them both- "If my wife had caught me attempting to make love to yet another woman... " (motions to slitting his throat) "oh, yeah. My beloved possesses a fiery temper." The first recorded use of the word "yeah" was not until 1902 so it would not have been used in 1744, the year in which Claire and Jamie are currently in.