How I'm Spending My Summer Vacation - S2-E17
Plot hole: When Kevin is upset about Paul leaving for summer and is just lounging around the house, he asks why they can't go on vacation. His mom says "Maybe next Summer." In the very next episode, which takes place in the same Summer, they are going on a "traditional" Summer vacation.
Scenes from a Barbecue - S3-E24
Plot hole: All the kids and Arnie are about to play a basketball game. Darlene and Chuck Jr. battle in Rock, Paper, Scissors to see who picks first. They both throw scissors, and then Chuck Jr. says he gets to pick first. They should have done it over, since that would be considered 'a tie', he really didn't win. (00:12:10)
Plot hole: The mysterious neurosurgeon Dr. Quinn was mentioned by the fake Myles Taggart under the pretence that his presence was announced by the local newspaper. It turns out that his invitation by the university was a lie - presumably no newspaper story existed at all then, or the university would have acted on that. Taggart is exposed as a fraud, Quinn's link to him is so direct that he was even the one who invited him (instead of goading the Colonel into inviting him like it was his own idea) and yet the Colonel continues to trust Quinn blindly - he even has the whole night and day to think about this simple fact, nothing was decided on the spur of the moment.