The sign by Mystic Falls cemetery says it was established in 1792, but later episodes state the town was built in the 1860s
In the first few episodes, Damon is shown to have unique powers (such as controlling the fog and crows, showing Caroline visions from far distances, etc). These powers are not shown again, and it is not explained how he was able to do this.
The set for Jeremy's room has a crokinole board placed on the wall, like a dart board. Crokinole is a game that must be played on a flat, horizontal surface, and the boards are usually heavy as they're made of wood, so hanging a board on the wall is sort of ludicrous.
In the state of Virginia (where Mystic Falls is said to be) if you are under the age of 21, all IDs are supposed to be vertical. However, when Sheriff Forbes looks in Vicki's purse her ID is horizontal.
It is possible that Vicki had been using a fake ID, but wouldn't the sheriff notice it was fake, especially when she knew the Donovan family personally, and knew Vicki was underage?
In all the other times that either Damon or Stefan have been cut, the wound healed almost instantly. However when Damon pierced his own skin to force Vicki to feed, the wound stayed open for the entire time she was drinking from it.
Logan, the newly turned vampire, claims he cannot get in his own house because the fact that he lived alone meant that there was no one to invite him into his own home. he got no one to let him in as Logan lives alone. However, later in the season, Samantha Gibbons was killed by Damon Salvatore after she had been compelled to refuse to give him an invitation, thereby allowing him to enter the house without it.
It is far more likely that Damon would have smelled the pile of bodies in the Fell Warehouse before Logan pointed them out, especially with his heightened senses.
During flashbacks a copy of Renoir's "Two Young Girls at the Piano" can be seen on the wall behind Katherine. This painting was painted in 1892, which happened 28 years after the flashbacks took place.
The men of Mystic Falls should be fighting in the Civil War, as they seem able-bodied enough.
Giuseppe's tombstone says that he died in January 1864. However, the Founder's Ball, during which he was alive, took place in September that year.
If Johnathan Gilbert found out that Pearl, the woman he loved, was a vampire who could walk in the sun, wouldn't he write that down in his journal?
Jenna's search changes from "Peterson" to "Trudie Peterson" from one frame to the next.
Why did Isobel have Trudie killed? If it was so she could not tell Elena too much, why not just kill her earlier before she could say anything at all?
When Matt and Caroline were watching TV on the couch, whenever the camera cuts to Matt, his arm is leaning on the couch behind Caroline. Although, whenever the camera cuts to Caroline, Matt's arm is around her shoulder.
When Elena pulls the stake out of Harper's leg for Stefan, when they are rescuing Stefan, Stefan's hands are still tied and the rope is trailing over his shoulder, then Damon tells them to hurry up and they are leaving the room and Stefan's hands are untied and the rope nowhere to be seen.
Damon can enter the house after killing Samantha Gibbons. However, Logan could not enter his own house after becoming a vampire.
Stefan tells Elena he had to learn to control his bloodlust on his own, but in The Dinner Party it is shown that Lexi helped him control it.
In History Repeating, Stefan says that he saw Katherine get dragged inside the church; however, in Blood Brothers, it is revealed he was unconscious due to his vampire transition at the time and only Damon saw her get dragged inside.
While it is possible that Grayson, as a doctor, could fake a birth certificate that claimed Elena was his biological daughter, Miranda could not have faked a pregnancy. Someone was bound to have noticed she had apparently given birth without warning. In continuity, they may have claimed they were hiding the pregnancy.
At the end of Fool Me Once, Pearl tells Damon that Katherine made a deal with a guard who was obsessed with her and that he let her go, but in Memory Lane, it is revealed however that she made a deal with George Lockwood for the moonstone to set her free. It is possible that George used this guard as part of the deal.
The way the fight scenes are presented often don't make sense power level wise.