Inconsistencies

Season 2, episode 12 involves a phonograph or "talking machine." Whether it was invented yet depends on your point of view. The man selling it says that Thomas Edison invented it the previous year. Edison invented the phonograph in 1877, implying this is 1878, which was also implied in the Season 1 episode "Survival," so yes it was invented. But in a later episode it is only 1876, so no it wasn't.

The term "airplane" is used when it should have been "flying machine."

Pa is hammering in round nails, which weren't in use in the United States until the 1890s.

In Season 2, Episode 16, "The Runaway Caboose," Mr. Edwards worries about his son Carl missing school. In the very next episode, the Edwards children are not in school. They are also absent from school and church in a number of other episodes.

In Season 2, Episode 21, "Soldier's Return," Mr. Whipple is a veteran of the Battle of Shiloh, which it is mentioned several times took place 12 years earlier. The previous episode took place in early July 1876, so the Battle of Shiloh, which took place in April of 1862, should have been 14 years earlier. Also (SPOILERS) Mr. Whipple ran away from the battle on two good legs and hid until it was over. He received a medal for having a severely wounded leg, bad enough to have been treated by a surgeon, have a limp years later, and result in a morphine addiction. It was never explained how he got the bad leg.

EDIT: I think I figured out what happened. He would have been executed for cowardice if it was known he ran, so after the battle he found some dead soldier in an out of the way place, took his gun (remember he didn't have one but only a bugle) and shot himself in the leg, then waited for help (if he couldn't walk) or went to the nearest field hospital (if he could). This was just both too gruesome and too long to be shown.

In Season 3, episode 2, "Bunny," Bunny was male in "Christmas at Plum Creek" up until Mr. Oleson said, "Atta Girl," at which time she became female. She is female in this episode.

When people collect in Nellie's room to see if she is all right, a shadow moves in the lower right corner of the screen as if someone suddenly moved out of the shot.

When Nellie breaks the glass on a sampler of a doll, the sampler seems to read "Porcelain 1920," though the doll looks older style than 1920, but the date sure looks like that.

In Season 3, episode 3, "The Race," Bunny the horse is still female, although Charles refers to the horse not being able to pull "his" weight around the farm.

Mrs. Oleson buys Nellie a horse against Nels's wishes and points out that the money came from her own bank account. Married women in America were not allowed to have bank accounts until the 1960s and many still had to have a signature from their husbands until 1974, which was only two years before this episode was made.

In Season 3, Episode 4, "Little Girl Lost," Willie ran into the church/school building to ring the bell when the rope was clearly on the outside of the building.

In Season 3, episode 7, "Journey in the Spring, Part 2," Bunny the horse is still a girl although Grandpa Ingalls says of Bunny, "She loves him," showing Bunny to be gender fluid to the last.

Charles uses the term "living room," which didn't come into use until at least the 1890s. The older term was parlor which became too associated with death, hence "living room."

Not so much an inconsistency, but a difference. The students of Walnut Grove School must have learned a lesson. In "Injun Kid," they practically killed the Native American boy, where in "The Wisdom of Solomon" their response to the black student was merely an unenthusiastic greeting.

Also in "The Wisdom of Solomon," when Solomon gets out of the bath it is visible that he is wearing shorts.

In Season 3 episode 22, towards the end in the scene in church, a child is seen wearing a very modern-looking sweatshirt.

In Season 4 episode 1, when Bandit jumps on the wagon there is a big space between the boxes for him to jump into. In the next shot the boxes are stacked close together completely filling the wagon and looking too tight to shift. In the next shot they have shifted to a different position. Also it is unclear how Kezia knew Bandit's name when all Laura said was that he was a stray.

Season 4, episode 7: when a patient is talking to Doc Baker in his office, a shadow which looks like a microphone swings over his head and back.

Also not exactly an error but for some reason Laura is neither seen nor mentioned. (It's been said that Melissa Gilbert claimed to have not been in only one episode but it was more like fourteen.)

Season 4, episode 8: Mary accidentally calls one of her employers "Mr. Dobbs" instead of "Hobbs." She was supposed to have been working for them for some time and should have known their names by then.

The year is twice identified as 1876, to coincide with the great Northfield, Minnesota, James brothers raid, though the episode is from 1977 and they did an 1876 episode in 1976 meaning they stayed in 1876 a long time, which I guess is allowed.