The Kidney Stays in the Picture - S7-E16
Plot hole: It is explicitly stated in this episode that Hayley was conceived in 1996. While normal for shows without character aging to change birth years to benefit the plot, this move was premature. Hayley's permanent age is stated as 18, and this episode implies that in 2012 she would be much younger than this. The whole plot centres around her being legally able to drink in a bar, so this is a pretty big mistake.
Plot hole: When Robin and Lily find Barney's secret Robin notebook, Barney and Ted are at the class, so Barney should have the notebook with him.
Suggested correction: This is a character mistake at best, and quite a common one. Students constantly forget their textbooks, homework, notes, supplies etc.
It should be noted that the plot hole isn't talking about "students." This isn't about forgetting textbooks, supplies, or homework. Barney is trying to be a better boyfriend to Robin and wants to learn Ted's secrets since Ted had dated Robin. If Barney has been writing everything down before, the mistake is pointing out why isn't he writing it down this time, except to allow the writers to have Lily and Robin find the notebook and the class.
Plot hole: When Jim's bank calls his office, Kevin is there and ends up pretending to be Jim. The Banker automatically assumes that he is not in Puerto Rico only from asking his address and social security number. It would have been very plausible for him to have forwarded the number to his cell phone, where the same scenario could conceivably occur (since The Banker refused to listen when Kevin tried to correct her).
Plot hole: If Andy really did enlist into the Army Reserves and "forgot about it" like he claimed a warrant for his arrest for desertion would have been issued, which the police would have seen when he was arrested for pot possession in the episode "Lude Awakening." After seeing the warrant, the police would have called the Army, and he would have been returned to military control to face desertion charges. He would not have been released and sent a letter a few weeks later simply telling him to report for duty or face military prison; the option to report for duty would have no longer been available when he was gone long enough to get put in deserter status.
Plot hole: There is no way that Zoey could've seen those boys hanging Chase's bike from a tree. She was way far from the window to even recognize his bike, and when Chase looks through the window, the bike was just now being lifted from the ground.
Mac is a Serial Killer - S3-E10
Plot hole: Early in the episode, it is mentioned that the serial killer is murdering "Young, attractive blondes", yet at the end of the episode, when the gang finds all the severed heads in the killer's refrigerator, almost none of them are blonde or young, contradicting what had been previously established.
Plot hole: The whole episode is based on false facts. Cody was getting an A in woodwork and Zack was failing in all his classes the previous year at school (according to their report cards in Ep "Smarter and Smarter"). So Cody knows a lot more about woodwork than Zack and should be able to make a clock, never mind a birdhouse, and Zack should not be able to make a clock.