The Adventures of Brisco County, Jr.: The entire series. For a series taking place in 1893 we have such things as rockets and rocket powered rail cars, functional tanks, and in one instance a Zeppelin. This show also provides an interesting explanation for the acronym U.F.O. as an Unearthed Foreign Object.
Angel: A flashback to the early life of vampire Darla occurs, according to the caption, in the Virginia Colony in 1609. As anyone who's attended Virginia public schools knows, in 1609 the colony consisted solely of the struggling Jamestown settlement, which, in 1609-1610, went through what's known as the Starving Time and was almost abandoned. Yes, by 1609 the colony included a few women, but it certainly wouldn't have had an Olde English style inn to shelter Darla while on her deathbed waiting for a vampire master to drop in and turn her. All the writers had to do, to make the chronology plausible, would have been to add a couple of decades and set the flashback in, say, 1629.
The Big Bang Theory: One episode is bookended by the gang coming from/going to a Renaissance Fair, with Sheldon commenting in the intro: "Worst. Renaissance. Fair. Ever." and then listing all the common anachronisms relevant to Fairs... and, on the outro, the gang returns to a Fair, Sheldon now wearing Spock's uniform (including ears), holding a tricorder, and commenting as if he were on Star Trek, visiting another planet. (Could be a reference to a mid-nineties UK national LARP meetup, where one of the players spent a day playing just such a character.)
Blackadder: While it gets the eras correct (there is a firm difference between the Middle Ages and the Renaissance) it moves around in each era quite erratically. This is most notable in Blackadder The Third, which is ostensibly set during the 1810s, but features Samuel Johnson working in his dictionary (1750s) in one episode, the French Revolution (1790s) in another, and the Napoleonic Wars in the Season Finale (as well as Nelson featuring in Blackadder's Christmas Carol alongside George). Arguably, this is Rule of Funny-based Anachronism Stew.
Said Season Finale also features a mention of The Prince and the Pauper. The author wasn't even born until five years after the character mentioning the story died in real life. William Pitt the Younger died in 1806, yet features in the first episode. The Duke of Wellington features - he didn't become Duke until 1814, and only met Nelson in 1805, when he was major-general at the age of 36. The battle of Trafalgar (1805) hasn't happened yet. The battle of Waterloo was ten years after Trafalgar. And the Peninsula war, which was 1808-1814. Wellington didn't become duke until said war was over.
In-universe examples also turn up, as in series 1's "The Archbishop". Among the relics ostensibly crafted by Jesus during his early life as a carpenter are a cigar stand and a crucifix.
The Zippo lighter is featured prominently, particularly by Ben Hawkins who often uses his Zippo like a flashlight. The Zippo wasn't patented until 1936 and didn't gain popularity until WWII.
As IMDB points out, Sofie uses a mop, which wasn't invented until nearly 20 years later.
Also from IMDB, the show features beds with fitted sheets, an invention that didn't come about until later.
The attitudes of the female characters are more indicative of modern women than women in the 1930s. At the same time, the men who interact with the women simply accept the attitudes of the women.
Community: Played for Laughs in the Season 3 Christmas episode as Troy and Abed get Pierce to join the glee club with their song "Baby Boomer Santa" that reconstituted the last sixty years with lyrics like "And when the Commies gave the polio to Doris Day - Santa helped the Beatles chase McCarthy away!"
Danger 5: Features the eponymous Five-Man Band fighting Hitler and his Nazis in the 60s, in a setting which includes robots and dinosaurs.
Daniel Boone: The series (19641970) veered wildly across time, with episodes set before, during, and after the American Revolution (one in 1805/06) all mingled together.
Doctor Who: Think Hartnell stories with bouffant Sixties hair in ancient Rome or the French Revolution or the Middle Ages. Most of the time, of course, the series inverts this with alien-generated anachronisms (such as the Meddling Monk's gramophone records, tobacco and marker pens in medieval England).
The Doctor likes to enforce this trope himself sometimes, in order to impress people. Why would you have Roman Centurions and Victorian Reptile Women battling space-monks armed with lightning swords? Because its cool.
Happens again in "The Wedding Of River Song". This taking place in a 'modern day' London where Charles Dickens shows up on morning talk shows, pterodactyls are the equivalent of pigeons, Roman soldiers ride the tube, and everyone takes balloon cars to work. In this case, all of time collapsed into a specific fixed point; all of history was now occurring at once but all the clocks stood still.
Happy Days: Ended up like this. The first two seasons tried fairly hard to stay true to the '50s setting, but as the show became more popular, the producers started putting in references to trends of the '70s and '80s, and the actors started letting their hair and clothes look more contemporary.
Heroes: In season 2, Hiro Nakamura went back in time to 1671, where he meets a wandering English samurai. 1671 is well into the isolationist period in Japan, when any foreigner would be arrested and killed by local authorities. Granted, at first it seemed he always wore a mask to prevent people from knowing his true identity, but this was soon forgotten.
Highlander: With its historical flashbacks practically every episode, contains too many examples to list.
Hogan's Heroes: The producers blew their entire costume budget on reproduction German uniforms, so minor cast members had to bring their own gear. This means that anyone not a P.O.W. or in German uniform will be dressed in contemporary fashions of the late 1960s and early 1970s.
Honey, I Shrunk the Kids (the TV Series): Amy travels back in time to 1976 to date her history teacher. One scene shows an arcade, the Pong machine is not an anachronism, but in the background you can clearly hear people playing Asteroids, Donkey Kong, and Pac-Man; also they want to play Space Invaders. Space Invaders did not come out until 1978, Asteroids in 1980, and Donkey Kong and Pac-Man in 1981.
Jeeves and Wooster: Set in an idealized version of England at an indeterminate point between the World Wars, and largely picks and chooses on matters of detail — Bertie Wooster drives a mid-'30s car, for instance, but Prohibition is still alive and well when he visits the US, and perhaps more jarringly, the establishing shots of the city use Stock Footage in which the World Trade Center (finished in 1973) is clearly visible.
Kamen no Ninja Akakage: Set in Warring States Japan, the main characters use high-tech weaponry like flying machines and guns to fight kaiju.
Kings: Being a modern retelling of the story of the Biblical David (who is believed to have lived c. 11th century BCE) is already loaded with kinda-sorta anachronisms, but a particularly interesting one takes place in the fifth episode (sixth if you count the two-hour pilot as two eps), Judgement Day, wherein Jack (King Silas's son) makes reference to "cutting babies in half" (i.e. Solomonic wisdom to decide disputes). Solomon was David's son.
In the episode, "Meet You at the Fair," a modern hot air balloon is shown, complete with FAA-regulated numbers (N4011A), a system that didn't come into play until the 1940s.
In another episode, Harriet refuses an offer to buy what was implied to be Kentucky Fried Chicken franchise from Colonel Sanders, a man who didn't start his restaurant franchises until the 1950s.
The liberated attitudes of the women in the show, and the acceptance of these attitudes by the men.
In one episode, a girl says she was the captain of the girls' basketball team in her school. This is 20 years before the sport was invented.
Another episode has Charles tell Mary he has a peanut butter and jelly sandwich and that she ate them when she was little. Peanut butter wasn't to be invented for another 20 years.
One character talks about having a Waldorf salad. This was 10 years or more before the Waldorf salad was invented.
Catalogues seen in the general store didn't come into use for 20 years after the events of the show were supposed to have taken place.
In a trip to San Francisco to see the ocean, Charles gets a ride from a man who is supposed to be William Randolph Hearst. He was born in 1863 and would have been a boy at the time the scene took place.
Mad Men is usually very good for keeping the timeline straight, but in one episode, a hard-bound copy of "The Corps" by W.E.B. Griffin can be seen in Don's office. The book wasn't published until 1986.
Maid Marian and Her Merry Men: Virtually everything that happens. Despite being set in The Dung Ages, it still manages to have a Rastafarian (identified as such), a telethon, and sell-by date laws. Amongst many other examples.
Mash: A frequent offender. The series has an inconsistent timeline jumping back and forth during the Korean War (June 1950 - July 1953), but some egregious examples include:
A pinball machine in the officer's club that wasn't produced until the 1970s.
Pictures of the UH-1 Huey in the officer's club. The Huey began development in 1952, and did not see active service until after the Korean War.
One episode references Godzilla (first released in Japan as Gojira in 1954, and not released in the US as Godzilla King Of The Monsters until 1955), and another references The Blob, not released until 1958.
Radar, and sometimes other characters, occasionally do celebrity impressions (James Cagney, John Wayne, etc.) but quote lines from films made long after the Korean War.
One examples teeters precariously on the borderline between averting this trope and being another egregious example. In one scene, Charles brags to Hawkeye and BJ that he once had dinner with Audrey Hepburn. Hawkeye and BJ scoff at the idea that such a famous beauty would have anything to do with Charles...until he shows off a photograph of the two of them together. The problem is that Hepburn didn't become a truly major star until the release of Roman Holiday, a film that came out exactly one month after the Korean War ended. However, she did have some measure of fame during the War. She starred in the original Broadway production of Gigi from 1951-52. So it's at least possible that Hawkeye and BJ had heard of her during the War, but given where they're from (Hawkeye's from Crabapple Cove, Maine, while BJ's from Mill Valley, California) it's extremely unlikely.
Within that same scene, Charles states he's never seen any of her movies. She had been in some movies by that time, but mostly in bit roles...with one big exception, where she had a major supporting role in the 1952 British film, The Secret People. Once again, possible but unlikely someone during the war would refer to her movies.
In the episode, "Army-Navy Game," while Hawkeye and Trapper are trying to defuse a bomb, Henry yells at them with a bullhorn. The bullhorn he's using wasn't invented until 1954, after the Korean War was already over.
In the episode Der Tag, this trope is melded with some abysmal continuity checking: In the opening scene, we see Radar sleeping, a comic book on his chest. The comic is a then-contemporary issue of The Avengers, showing the 1970s-era logo (the A with an arrow logo). The scene cuts to Frank talking to Hot Lips on the phone. When the camera cuts back to Radar, he still has an issue of The Avengers on his chest, only now it's a completely different issue, with the older 1960s-era logo!
Moonlighting: The episode "Atomic Shakespeare", while set at a semi-Shakespearean, Semi-European time & place, had Bruce Willis as David as Petruchio on a horse with the BMW logos on its saddle blanket. Oh, and it also had Ninjas.
MythQuest: Has a few in its depictions of mythology:
Orpheus goes down into Hades to rescue Eurydice and plays with his lyre... "Greensleeves". From 17th-century England.
Blodeuwedd, a character from early Welsh mythology, appears in a large stone castle and is given a modern trial, including a lawyer.
Professional Wrestling can have a good deal of this, primarily because the programming is identified even in-universe as staged entertainment and doesn't have to correspond at all to reality. Much of it is in the spirit of Rule of Funny (the '50s greasers Deuce & Domino, two medieval Scottish warriors called the Highlanders, etc.), but occasionally it's done apparently without irony. The best example of non-ironic anachronism in WWE is probably Dolph Ziggler, whose hair and clothes give the impression that somebody puked every imaginable 1987 stereotype into a blender and then made a man out of it - and Ziggler is talked up by everyone as a straight-out arrogant "cool" guy rather than the laughable Disco Dan he really is.
Revolution: Not so much with the gear, which is par for the course considering the apocalypse, but the combination of a pre-industrial, feudal agrarian society and industrialized armies seems unlikely and ultimately unsustainable. Surprisingly Truth in Television; many societies had this kind of imbalance.
Rome: This series is generally very, very good in its depiction of Ancient Roman life. However, many of the suits of armor the characters wear would have been bleeding-edge technology in the year the film is set (assuming any examples existed at all).
Stargate SG-1: Justified Trope. A lot of the planets that they travel to, especially in the early seasons, are based off Earth civilizations (which, naturally, have not evolved at all in the centuries or sometimes even millennia that have passed, and all of which inexplicably speak English). Some, though, have changed a little bit, so we often see medieval-like cities with spaceships and teleporters. This always makes sense in context, but is still noticeable.
As Mitchell points out in one episode, referring to an Arthurian stash of gold with a clearly alien device hidden in it: "Which one of these things is not like the others?"
Possibly the best example is the entire episode Camelot from the end of the 9th season. Within about 10 minutes we go from Mitchell having a sword fight with a knight in armour to one of the most epic space battles in the entire franchise.
The Tudors: Consistently featured women wearing long, billowy sleeves that covered their hands. While this technically was Tudor-era fashion, these sleeves did not come into fashion until Queen Anne Boleyn wore them. Therefore, episodes that take place before Anne ascended the throne are historically questionable.
Underbelly: The pilot is set in 1995 and features the Spiderbait song "Shazam" (1999) playing on a stereo.
Wonder Woman: Filmed in the 70s. The first season was set in the 1940s. The anachronisms are abundant, but subtle:
A black Marine corporal is seen in Washington DC before there were any black Marine corporals.
Steve Trevor wears military ribbons on his uniform before they've been issued and before he would have had a chance to earn them. Later in the first season, after he would have had time to earn them, he stops wearing them.
Xena and Hercules: In their Universal TV series, live in a world where not only are all myths and legends true, but are also all happening within a few seasons of each other. The Argonauts sailed just a few years before Julius Caesar ruled, and Hercules was old friends with Vlad Tepes.
The producers explained early on that they were perfectly aware of this and did it simply to add to the camp value, further explaining the one rule they had was that anything BC was fair game, and AD was off limits. They missed the boat on that rule a lot: Boudica's rebellion in England (60AD), and Vlad Tepes/Dracula, who was a medieval ruler, Genghis Khan and his three sons, and the episodes in the modern world. King Arthur manages to get around the rule thanks to Merlin using magical Time Travel.
They also often show inventions that haven't been invented yet, ala The Flintstones. One episode had Hercules playing basketball. Another involved a giant spiderwoman, which lead to a "website" quip.
Nothing quite compared to the episode where Xena and Gabrielle are joined by a teenage Greek bard named Homer, who proceeds to tell them the story of... Spartacus.
Almost as bad was the episode featuring Hippocrates and Galen, with Galen being the older of the two. (Galen was born in 129 CE, about 500 years after Hippocrates died.)
Both Gabrielle and Aeolus are present at the birth of Jesus, and don't run into each other. Gabrielle also considers running off with a foreign poet named David (who's writing psalms as he and his friends travel), until she finds out he's married.
The Family Channel Zorro series, ostensibly set in Spanish California (no later than the early 1820s) was riddled with them: Sgt. Garcia, in a Christmas episode, remembers asking Santa Claus for presents in his childhood; St. Nicholas (not yet "Santa Claus") didn't become known even to east coast Americans of English background until slightly after that. Women serve on a jury. A young woman modeled on Annie Oakley mentions that she learned sharpshooting in her parents' Wild West Show, a form of entertainment not invented until the late 19th century.