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.
Attila: During their battles with the Huns and various Germanic tribes, the Roman armies are wearing armor that is roughly three centuries out of date.
Better Call Saul is supposed to be set in 2002, but New Mexico Rail Runner trains (which didn't start service until 2006) appear in several shots, Tuco uses a revolver model that wasn't made until 2010, car models from 2005 onwards creep into the background of several shots, and Jimmy's camera crew looks more like kids from the 2010's than 2002.
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. This is of course deliberate and lampshaded; tired of Sheldon's complaining, the rest of the gang told him he could pretend to be a Star Trek character investigating on the Planet Of The Week. Many of which did feature analogues to Earth history, where any "anachronisms" could be explained away as the planet's natives copying a period from Earth inexactly.
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.
Black Sails: The show plays fast and loose with the time periods of the various famous pirates it references and portrays as characters.
The Knights of Byzantium in Season 5 are an ancient Knight Templar order who use medieval arms and armour for no apparent reason than to have a cool scene involving Buffy fighting knights on top of a moving Winnebago.
The Trio in Season 6 combine magic with high technology to carry out their capers.
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.
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.
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).
In "The Time Monster", the Master sends an armoured medieval knight and a platoon of roundheads to do battle with UNIT soldiers in 1970s Britain.
In-Universe: In "Carnival of Monsters", the Doctor and Jo travel to what appears to be a ship from 1928. Only every twenty minutes, it gets attacked by a plesiosaur. The Doctor deduces from this that he's in a miniscope, a sort of futuristic pier attraction, assembled as a tableau by aliens who really didn't care whether or not the 'Tellurian' creatures in their show all existed at the same time in reality (and realised that watching humans running away from giant lizards is fairly entertaining).
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. Not to mention that intercontinental steam trains have taken the role of airplanes. 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.
As pointed out in the DVD Commentary, the tapestry on the wall behind Viserys during his introduction in "Winter is Coming" shows two ships blasting away at each other with cannons.
Despite the series' well-established Conlang, the slogan "Kill the Masters" is written in English language and script in both "Oathkeeper" and "The House of Black and White" when Meereenese should be writing Low Valyrian in glyphs similar to those Talisa uses in Season 3. This is likely Rule of Perception, since even characters who speak entirely in Conlang can read it effortlessly.
Done deliberately in the sitcom The Goldbergs, set in an intentionally unspecified part of the 1980's. Justified in that all of the stories are presented as childhood memories in the mind of the show's protagonist, who admits to being an Unreliable Narrator.
Technology is all over the map. Most cars date from the 1970s, characters wield modern firearms while carrying old flip phones, televisions are old CRT models, the dominant portable music medium is cassette tapes. There's an old-fashioned glass aspirin bottle without a childproof top is seen in the pilot. Typewriters are used instead of personal computers in the first season, and those computers that do show up are old desktop boxes with CRT monitors... except in the future Batcave where Thomas Wayne had set up an office that had flat-panel displays and a server stack.
World events are hard to pin down or contradictory. While Gordon is stated to be a war hero, there's no information about which war he fought in. A radio quiz states that there are 118 known elements, which has only been true since 2010.
The culture of Gotham confirms to no one specific time period. Fashions run the gamut from old-fashioned to modern to entirely original. Homosexuality is as tolerated as it is in modern times.
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.
In one episode, Sgt Schultz is showing off the "unsinkable" Bismarck to the POWs, some of which were Americans. The Bismarck was sunk several months before the USA entered WWII.
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.
Also, in one of the New York episodes, you can clearly see a Coke sign using the slogan "It's The Real Thing", which Coke was definitely not using in the twenties or thirties. They didn't start using it in their advertising until the late sixties and early seventies.
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.
Laverne & Shirley: The show had little issue with using props available only in contemporary times. In one episode a character is eating cereal from a box of Wheaties featuring a famous photo of Bruce (now Caitlyn) Jenner winning the gold medal in the Olympic decathlon in 1976.
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.
The "Cavemen Pathologist" sketches has, as the name suggests, cavemen who are pathologists. Of course, being cavemen they lack the terminology or technology to do a crime scene reconstruction, only managing to figure their victim was killed "some time before now", and in the attempt wind up killing their only witness. Not to mention the part where the caveman constable mutters that the development of stone-based technology is a "double-edged sword", much to the confusion of him and his partner.
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.
Paranormal Witness(as well as several other paranormal documentary series in the same vein) have interesting anachronisms- most notably, an episode set in 2006 shows the characters being spotlighted in it as using smartphones. The iPhone came out in 2007.
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).
True to the books it's based on,A Series of Unfortunate Events (2017) has a field day with this. To start with, the Baudelaires seem to have vaguely 80's-ish wardrobes, Olaf mentions buying an hourglass online, Snicket mentions gay marriage being legalized (happened in 2015 in the U.S.), the Architeture is all over the place, all cars seem to come from the 40's, Mr. Poe dresses based in the 20's fashion, a group of people dresses in a 50's fashion shows up, and Sunny at one point suggests using Uber to get somewhere. And don't get us started on Bo Welch's eclectic set design...
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 season 2 finale of Supernatural takes place in Cold Oak, South Dakota. Given the hitching posts, wagon wheels, and old west appearance of Cold Oak, it appears to be mining town that was abandoned in the 1890's or earlier. Yet we see a "Push" sign on the door to the cobbler's shop, a Hart-Parr 18-36 tractor, and an upholstered quilted foam mattress in this ghost town.
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.
The Wonder Years has many anachronisms, especially with plot-relevant music being released later than the date the episode is supposed to take place in. The pilot takes place in 1968; Tommy James and the Shondells' "Crystal Blue Persuasion" was released in 1969, and the book "Our Bodies Ourselves" was published in 1973. "Alice in Autoland" is set in 1973; Johnny Rivers' "Swayin' to the Music (Slow Dancing)" wasn't released until 1977, and the plumbing fixtures are from the 80's-90's. "Scenes from a Wedding" takes place in 1972, and Jim Croce's "Bad, Bad, Leroy Brown" was released in 1973. "Heart of Darkness" takes place in 1968-1969; The Doors' "Riders on the Storm" was released in 1971.
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 who mentions being with war with the Ottoman Turks (you know... In Ancient Greece).
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.
Used intentionally in Westworld. The setting is a theme park based on the Wild West, so most guests likely won't notice. Among other things, the mechanical pianola plays songs like Soundgarden's "Black Hole Sun" and The Rolling Stones's "Paint it Black", while Dr. Ford mentions that he included some Gertrude Stein references in an old cannibal cult leaders' quotes for fun.