At the auction for the captain's journal of the "Black Rock" the auctioneer says 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 1970s portions, all of the spectacle wearing characters have an anti-reflective coating on their lenses (the greenish hued reflection). In the 1970s these were generally reserved for optical equipment lenses.
The two Korean characters are married and share the family name "Kwon", and the identical family name becomes an important plot point in the final season. But when Koreans marry the wife never takes her husband's family name. In fact in Korea it is not permissible even to marry someone with the same family name except in rare circumstances.
James "Sawyer" Ford is a career con man with an extensive criminal record, which is covered throughout the series. However, everybody except New Zealanders must get a visa to travel to Australia, and Sawyer never would have been approved. The "character requirement" prevents people with substantial criminal records or known criminal associations from entering Australia.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
During the beginning of the series, mainly the first season, when the airplane crash is regularly highlighted, we are informed in a few of the episodes that Rose's husband Bernard was visiting the lavatory. However, during Lost: The Other 48 Days Bernard is found stuck in a tree buckled into a seat and sitting next to a deceased passenger. However, Bernard could have been outside the lavatory and next to an empty seat when the plane hit heavy turbulence and quickly buckled himself in before the plane broke apart.