In season 1, Charlie says to Jack that he can't swim, yet in season 3's 'Greatest Hits' it shows that his dad taught him how to swim when he was young. He also swims to the Looking Glass station with no problem.
At the auction for the "Black Rock"s journal the said it sailed and disappeared in 1845, 20 years before Alfred Nobel applied for his patent for dynamite. Yet the ship supplied dynamite through the series.
During the scenes in the 1970s, all of the spectacle wearing characters have an anti-reflective coating on their lenses (the greenish hued reflection). While these coatings are very common today, in the 1970s they were generally reserved for optical equipment lenses.
In a flashback it is shown that the character Sun Kwon had no English ability, but learned English over a period of several months. Regardless of the amount of effort it is absolutely impossible to reach her native-level ability in such a short amount of time.