When Jo wears the black dress, her left arm doesn't have the skull and bone ranger tattoo which was clearly visible on her arm in the fist episode.
While looking over the papers from Fargo's office to find some answers about the missile, Henry's jumpsuit initially doesn't have a patch to identify his current position or duty. A few seconds later, the "Mechanic" patch is on his jumpsuit.
Allison said the test chimpanzee's autopsy was inconclusive. The correct term, that a brilliant medical doctor should know, is necropsy.
During the mock Nobel Prize ceremony, Ed Quinn says ".. a man whom the Nobel committee and the Norwegian parliament selected for this year's award of Nobel Prize..." which seems to be incorrect, as it is Swedish-selected groups (which are separate from the Swedish parliament) that decide the winner for the Nobel prizes in most fields. However, the Peace Prize winner is determined by a group selected by the Norwegian parliament. It is stated that Dr. Thatcher lost out to Linus Pauling, who did indeed win the 1962 Peace Prize.
Thatcher was immensely disappointed when Pauling won his second Nobel prize. But Pauling's second prize was for peace, not science, so he would not have competed with Thatcher.