In the opening shot, when the last of the five remaining Dutch vessels sails the Pacific, it carries the Dutch flag, red stripe on top, white in the middle and bottom's blue. This is "in the year of our Lord one thousand five hundred and ninety-eight..." but at that time, the top stripe on the Prince's Flag was orange as it first appeared in the 1570s. It was changed to red in the mid-17th century but it took until 1796, almost a century after the series is supposed to be set, before red white and blue were officially announced to be the official colors.
After the storm at sea, Rodrigues is recovering and tells Blackthorn, "There's a bottle of grog over there. Bring it to me." The term "grog" came from British Admiral Edward Vernon, for whom Washington's estate "Mount Vernon" was named. Vernon habitually wore a coat made of "grogham," a coarse cloth of the 17th Century. He was referred to by his men as "Old Grogham." Vernon disapproved of the British Navy's tradition of giving the sailors a daily ration of rum. He watered down the rum that was passed out to the men. The men derisively referred to this watered rum ration as "grog." Thus the term "grog" did not enter the language until the 1760s and would not have been used in 1600.
Anjin-san receives a flintlock pistol as a gift from Lord Toranaga. Flintlock pistols were not in wide use until 1630, thirty years after the show's setting. The wheel lock and match lock pistols and muskets, also used in the show, are correct. Additionally, the Europeans were reluctant to export the latest firearms to Japan, preferring to keep the superior weaponry to themselves.
Blackthorne says to Toranaga that Portugal and Spain had shared the world between them through the "Treaty of Zaragoza". In fact that division was made by the "Treaty of Tordesilhas" (1494), signed by King João II of Portugal and King Fernando and Queen Isabel of Spain. The "Treaty of Zaragoza" was signed by King John III of Portugal and Emperor Carlos V in 1529 and its only aim was to clarify the question about the control over the Maluki Islands.
In the sequence where Mariko is to commit Seppuku, one of the camera lenses was dirty. This is clear in four moments, with the camera showing a black dot over the blue sky on the right of side of Mariko's face.
During the earthquake scene, when two of the "crevasses" open up to drop samurai into them, you can see the wooden planks used to loosely cover the crevasses so people could walk on them until their supports were blown, hidden by a thick layer of dirt.
The goof item below may give away important plot points.
After Mariko's death, Blackthorne is led back to his galley by Brother Michael and Father Dell'Aqua. It is daytime when they leave the Jesuit Mission, but in the next scene when they get to the "Inquisition" staged by Captain Ferriera, they are carrying torches and it is fully dark.