Throughout the series, children seem to come and go in the school, but the school stays the same size, but they act like the new characters were there all the time. Also some characters only appear once, are said to have stayed, but disappear.
Caroline Ingalls is seen supposedly spinning wool into yarn on a spinning wheel, but the family did not own any sheep.
Charles and family were saying goodbye to Mr. Edwards. As Mr. Edwards is walking behind the Ingalls' wagon to leave, you can see a crew member moving out of the way from the other side of the wagon. It's quick, so you have to look for it.
Edit: It's actually the foal attached to the wagon, you can see it when they on out and the wagon is pulling away.
At 8:20-8:35, when the Ingalls family stops at the river they have to cross, the camera angle pans upward from the ground to show Ma and Pa who are sitting in the front of the wagon. In doing so, an airplane contrail can be seen in the sky above them.
When the family gets to their new "homestead" there isn't a tree in sight. Suddenly there's enough wood to build a house and barn.
Ma is shown sitting at her spinning wheel seemingly pulling yarn from a bundle of fluff rotating on the wheel's bobbin. This would not produce yarn. On a spinning wheel, yarn collects on the bobbin while the spinner carefully feeds small amounts of the wool into a flyer where it becomes twisted into yarn.
From 21:15 - 21:30 the camera pulls away from the wagon as the family heads towards the place they will make their home. As it does so, tracks can be seen on the ground.
In the first season, they mention Custer at Little Big Horn, which occurred in June, 1876, but do not have the Centenary til what seems like years later, which was July, 1876. Also they mention peanut butter, but its inventor, George Washington Carver, was only a boy at this time. In addition they more than once play Onward Christian Soldiers, and although the Hymn was written in 1876, the Music was not until 1881.
Pa Ingalls is seen riding a horse on grass but the sound of horse hooves on pavement can be heard.
After "Pa" brings home the cow, "Ma" exclaims, "...now we can have milk." However, the cow is clearly a bull.
Though indigenous men did walk into the Ingalls cabin and scare Ma half to death, there is no evidence that they fondled or physically threatened her.