Corrected entry: Right in the beginning, the show spells "Manhattan" wrong; they spelled it with only one "t". (00:00:30)
Corrected entry: When they go to the alternate universe (in part 1), they need Olivia (the one from this universe) to open a passage and the others to give her a boost, but when they come back, it's the alternate Olivia that is there, and in the first episode of the third season, they confirm that she doesn't have the ability to open a passage. So they should never have been able to come back with the wrong Olivia. So much of the show is based on "who" can travel between these alternate universes and "how", such a blatant mistake ruins it.
Correction: William Bell provides the energy this time and not alternate Olivia.
Correction: They came back to the "green-verse" with the power from William Bell. That's why Walter and William drove to Cambridge to get the machine from the old lab, they didn't think Olivia had the strength to do it on her own. That's why they were able to take back Faulivia with them. Without Bell they wouldn't have had the power for it to happen.
A New Day in the Old Town - S2-E1
Corrected entry: Early in the episode it is shown that the shifter's body changes to meet the victim. However, later in the episode it is shown to create a completely new body, leaving a total of two corpses, not one. (00:01:30 - 00:37:20)
Correction: There are two bodies in the basement, because the shapeshifter killed two people in the hospital: the nurse and Charlie. It hides Charlie's body under the towels (where he likely hid the nurse before) and makes the agents believe that the body of the nurse is the dead shapeshifter. After they've taken her away, he throws Charlie's body into the fire.
A New Day in the Old Town - S2-E1
Corrected entry: Peter tells Olivia that the Greek phrase she said on waking up was "Na einai kalytero anthropo apo ton patera tou" and translates this as "Be a better man than your father." The Greek is grammatically incorrect (it should be "kalyteros anthropos...") and Peter's translation is wrong. The translation of Olivia's phrase is "May he be A Better Human Being than his father." What Peter said is a translation of the Greek phrase "Na eisai kalyteros anthropos apo ton patera sou." (00:37:40 - 00:38:25)
Correction: His mother was making a wish. No one said it directly translates as anything. He says "It means." Referring to its meaning for him personally, to fulfill his mother's wish. Also she may have simply told him as a child a simple meaning for him to relate to. Knowing (Peter's mum) Elizabeth's character from later in the show, that seems very likely in its own right.
Correction: That scene takes place in the alternate universe, where the correct spelling is "Manhatan." Several later episodes use the same location name, and there are a few clues that this scene doesn't take place in our "Manhattan, " such as coffee being rare.