Saga of a Star World (1) - S1-E1
Plot hole: Apollo takes Adama down to the surface of Caprica aboard his Viper, yet the Viper is a single-seat fighter. This is the only time the single-seat craft is used to carry two people (though there is no shot of the two aboard - it is simply shown landed with the two characters nearby).
Suggested correction: It was 2 seater Viper training aircraft.
Plot hole: In the first season, Lucy Ewing had an affair with Ray, the farm hand at South Fork. Several years later, it was revealed that Ray was actually Jock Ewing's illegitimate son, making him Lucy's uncle. However, the subject of their affair was never raised, even though they surely would have been at least somewhat traumatized upon learning of the accidental incest.
Plot hole: When Danny Kendall first appears in series 9 (1986) he is in the third year, then the fourth year in series 10 (1987) but in series 11 (1988) he is put down a year to help him catch up with work he missed as he was in hospital for most of his fourth year. But in series 12 (1989) he is in the fifth year when he would be aged sixteen going on seventeen and he would not be legally obliged to stay on at school, as sixteen was the legal school leaving age then. When Danny died in series 12 the school planted a tree in his memory with a plaque in front of it giving his birth date as 8th January 1973 and death date as 22nd January 1989. His birth date should have been 8th January 1972, as he would have been seventeen years old when he died.
Plot hole: Michelle and the girls are winning at Baccarat every time due to a system where they know the cards in advance by seeing through them. However, all betting in that game is done before cards start coming out of the dealer's card chute onto the table. So even with their cheating system, they realistically couldn't always win, as the cards are not already face down on the table.