This is especially common for the occasional "What's Wrong With This Episode" episodes. Every now and then for April Fools' the show had a contest to give a prize to whoever could point out the most errors. These episodes are guaranteed to be more Big-Lipped Alligator Moment than actual episode as the characters turn into puppets, the show is shot entirely in The Sims, actors switch places, or a hundred other random occurrences.
There is also the "Drew Live" episodes which are essentially cross overs with Whose Line Is It Anyway? in which the entire episode is shot live and the actors will at any random point be forced to ad lib a new line, or break from the show entirely to play out a Whose Line is it Anyway? improv game.
Daffy Duck appeared in the season 3 finale looking for a job at Winfred-Louder, in the opening scene. After Daffy leaves, the incident is never brought up again. You'd think that if a classic cartoon came into the office, most people would be talking about it for hours on end!
A lot of the cold openings qualified. Another example is one where Drew's office is attacked by aliens. Turns out it's All Just a Dream that an alien Drew had.
The musical numbers generally come across as this, at least for people used to them. They follow the 1950s musical rules and not the 1990s sitcom rules.
Season 4's "Drug Co." is essentially a BLAM Episode where Drew and the gang break into Lewis' workplace to rescue his dog Speedy. They find a lot of strange things such as a human-like insect, holographic dobermans, a crystal ball that makes you smarter when you hold it, etc. The funny thing is, Drew and his friends don't seem very perplexed by any of these strange goings on (probably because they already know Drug Co. is a place where strange experiments are done), and after these events, they're never brought up again.
"Funny Aneurysm" Moment: in "Hush Little Baby" when Steve is leaving the house, Mimi throws him a suitcase, saying that she has it in case Steve ever cheated on her. Two seasons later Steve does end up cheating on Mimi.
In Season 2's "New York and Queens", Donald Trump has a brief cameo during the scene where Drew's truck breaks down, and Trump offers to help them. Kate seems rather excited about meeting him. This all can be a little jarring to anyone more familiar with Trump's current reputation.
A recurring element of the show is that Drew is overweight and okay with it, he has a good life and enjoys eating. The real Drew Carey revealed that he struggled for a long time hating how he looked, especially on camera, but after he took over The Price Is Right his brother died of a heart attack and that shook him into better eating habits and exercise.
Even though the sight of Lewis and Oswald dressed as Batman and Robin is already funny, it only got funnier when Diedrich Bader (Oswald) went on to voice Batman.
The jokes about Drew's weight gain have become this since Drew Carey's real-life drastic weight loss
The "Rocky Horror/Priscilla" dance-off from Season 2's "New York and Queens" shows Mr. Wick taking part in "Priscilla", and dismissing Rocky Horror. This is funnier once you find out that Wick's actor, Craig Ferguson, was in a 1991 stage production of Rocky Horror playing Brad Majors.
In Season 6's "Mimi's a Partner", Drew complains about having nothing to do while being unemployed.
"I'm sick of just sitting around watching game shows, so somebody COME ON DOWN, and kill me!"
Oswald and Lewis. Lewis had to enter marriage counseling with Oswald and his fiancee because he plays a bigger part in Oswald's life than Oswald's girlfriend.
In one episode where Lewis thinks he's dying from a cold, he decides to go to the Warsaw naked. Mr. Wick asks for Lewis' help moving his car, compliments Lewis on his skill with a razor, and then the two of them enter the Warsaw naked together.
Mr. Wick: Oh, thank you, Lewis! I've never felt so free!
Jerkass Woobie: Drew may not be the nicest guy on the show, but he certainly goes through a lot more crap than he deserves too. Mimi also has shades of this, but her Woobie moments are scarce.
Retroactive Recognition: A lot of actors who became famous on Scrubs had small bit roles on The Drew Carey Show beforehand. Sam Lloyd (Ted the Lawyer) appears as a delivery man in Season 3's "The Batmobile", Aloma Wright (Laverne Roberts) plays a security officer in "Good Vibrations", Nicole Sullivan (Jill Tracy) tricks Drew into marrying her in "Drew Gets Married", and Neil Flynn (The Janitor) was one of Kate's boyfriends in one episode. Not to mention Christa Miller (Jordan) starring in this show as Kate.
Seasonal Rot: The last two seasons which certainly felt like the showrunners were running out of ideas, often about Drew trying to get a girlfriend. This was not helped at all by the leaving of Kate's actress.
Strawman Has a Point: Mimi, in the pilot episode. Okay, she was way out of line calling Drew out during her job interview, accusing him of being sexist (which he wasn't), and making a scene in front of the other workers, but in Mimi's defense, Drew did not handle her interview very well. First of all, he makes a crack about her marital status (which Mimi actually ignored), and then has the gall to tactfully say to her "We WILL be interviewing a lot of people for this job", which is, as Mimi puts it, a tactful way of saying, "We're not hiring you." Any HR director will tell that you do not say anything like that to an applicant as it's very discouraging to them, and it makes the company look bad. Mimi is understandably pissed by this and deduces that if she were hot-looking he'd have hired her right away. Drew insists this isn't the case, except in the very next episode, we see him hiring a woman (Lisa) who is very attractive, and actively flirts with him, and eventually starts dating him! When watching the first two episodes, and seeing all this unfold, it's not hard to side with Mimi and think maybe she had a point.
Mr. Wick in his debut episode. Throughout the series Drew complains about not moving up the company ladder, and being stuck in a dead-end job as Director of Personnel. Wick point-blank tells Drew that he simply is not capable of handling a higher management position. If one watches several other episodes where Drew has to fire somebody, even for legitimate reasons (stealing, assaulting a customer, etc.), Drew will often try to find ways to keep said employee working at Winfred-Louder, or he'll be too intimidated to fire the person, usually out of guilt. Given that a manager's position requires someone with a bit more backbone, and the ability to tell when someone doesn't belong in a company, it's clear that Drew really doesn't deserve to be a manager (not that Wick does either as he's shown to be a whiney Jerk Ass).
Tear Jerker: The end of "The Bachelor Party" where Drew watches the sex tape he made with his fiancee and the audience learns it's an attempted sex tape where his fiancee realizes she's regained her old weight and breaks up with Drew, saying it's his fault she became fat because she "can't be around someone who loves to eat". At the end of the episode he rewinds it and watches it over and over. Made worse since her parents used this same phrase to describe Drew in an earlier episode but it was played for laughs.
Really, the fact that Drew's lovelife is such a trainwreck in the first place, given that he is a genuinely good, kind person who honestly attempts to put others first. Sometimes at the cost of his own happiness.
When Mr. Wick is caught with cocaine he's plays it off until Drew insist he's got to go to rehab. He sits down in his chair and with a weak broken voice mutters "I'm suddenly very tired." Even more so when you remember Craig Ferguson's real life struggle with drug addiction and alcoholism.