Calvinism and its associated sects are portrayed as self-righteous, morbid, and borderline misanthropic. Father Brown goes so far to say that Presbyterianism isn't really Christianity (in an ironic twist on No True Scotsman).
New religious movements are portrayed as faddish occupations for bourgeois people who want to be trendy and modern. If their leaders are present, they are either villains, Straw Hypocrites, or both.
Non-Abrahamic religions are given the Hollywood Voodoo treatment, as was typical for Europeans at the time. Chesterton admits that any "tribal" practices in his stories are made up and have little, if any, elements of truth to them, but they still play up stereotypes for all they are worth.
Hinduism and Buddhism are sometimes criticized as excessively nihilistic and world-denying. This one is a clear case of Writer on Board, as Chesterton makes a similar criticism in his non-fiction and apologetic works.
The Church of England is subtly mocked in some stories as an inferior version of Catholicism that stole a lot of the Roman Church's property in England. In "The Vampire of the Village," Father Brown pokes some fun at the fact that "England does not know much about the Church of England," referencing its rather complex history of schism and doctrinal differences. This comes off as more of an Affectionate Parody than the others, which isn't surprising given that many of Chesterton's close friends were Anglican (and he himself was at one point).
Accidental Innuendo: The Red Moon of Meru features a doctor peddling phrenology, the then-popular, now long-discredited practice of measuring intelligence by the curves of a person's head. He walks up to a fashionable lady and offers to "feel her bumps".
Fridge Logic: In the first story, when Flambeau asks for the package, he reveals that he actually swiped it some time ago in the next sentence. Why would he then stay with Father Brown and ask for it at all? It's possible that, being a Gentleman Thief and all, he simply wanted to enjoy The Reveal and mock the celibate dreaming simpleton Father Brown by outsmarting him. It would be out of character; for example, at one point he lists various types of people that can be profitably robbed or swindled, along with the appropriate setting for the crime in each case.
Harsher in Hindsight: In "The Honour of Israel Gow", a horrific incident of grave desecration turns out to have an innocent explanation. (A Literal-Minded and somewhat obsessive character who had been bequeathed all the gold in the deceased's possession realised after the funeral that he had forgotten about the dead man's gold tooth crown.) During the Holocaust, the Nazis would become notorious for going to similar lengths to plunder their victims' wealth.
Magnificent Bastard: M. Hercule Flambeau is an archcriminal and master thief, who is well known for being a Master of Disguise. First introduced in The Blue Cross, Flambeau befuddles the police to sneak into a convention of priests, disguising himself as one to steal a holy artifact. Only stopped by the genius of Father Brown, Flambeau accepts his defeat with grace and style but returns many times to pull off new brilliant heists until he ends up becoming a brilliant detective in his own right.
Once Acceptable Targets: Chesterton's willingness to capitalize on popular (in his day) stereotypes of Africans, Chinese people, Indians, and Jews have not aged well at all. See Values Dissonance, below.
Spoiled by the Format: The first mystery, "The Blue Cross", is told through the Sympathetic P.O.V. of the great French detective, Valentin. So when the unremarkable, seemingly bumbling priest (who isn't even named Father Brown untill near the end) solves the mystery, it would've been a huge shock to readers when it was first published in a magazine. But the most likely place for a modern reader to pick up this story isn't titled The Complete Detective Valentin Stories, so it ain't that much of a surprise.
Values Dissonance: Chesterton's racial and national attitudes were actually very moderate for the early twentieth century, but some will often strike a sour note for modern readers in the midst of his most enjoyable works, as for example in "The God of the Gongs." His religious views, on the other hand, were entirely conscious, and will strike the reader as either refreshingly forthright or offensively aggressive, according to taste. An article published at the Golden Age of Detective Fiction Forum, The Sins of the Saint: Racism in GK Chesterton, written by a Chesterton fan, analyzes 15 Father Brown tales that seem to contain this and absolves some of Unfortunate Implications and not others. It also points that a lot of classic authors of Detective Literature (Agatha Christie, McDonald, Burton Stevenson) also had racist views, and he asks the reader to take in mind the purpose of the work (they were not racist propaganda).
What an Idiot!: A lot of the stories involve people missing critical details that only Father Brown could spot. This gets taken to ridiculous lengths in a few stories:
"The Miracle of Moon Crescent" has everyone discussing a supernatural occurrence of a man vanishing from a high-rise apartment and reappearing garroted and hanging from a tree in a park just next to the apartment building. The police and witnesses are so caught up in what they witnessed that they miss the obvious part about the rope being long enough to reach from the park to the apartment.
"The Perishing of the Pendragons" has a seafaring scam dressing up a map of a river as an exotic land. Flambeau points out that not many people could have seen through it though.
"The Man in the Passage" has the witnesses to the crime giving different descriptions of a murderer. Turns out whom they saw was their own reflection, which is awkward and bordering on idiocy.
The TV series
Acceptable Targets: The death penalty. Father Brown's opposition to capital punishment is a major recurring theme in the series, and in many episodes the stakes are upped by the good Father having to find the real killer before the inevitably innocent person arrested by the bumbling police is sentenced to hang.
Alternative Character Interpretation: Is Mallory the least competent of the show's three inspectors, or the most self-aware? Valentine and Sullivan have cool, professional demeanors, but their self-confidence tends to make them into impenetrable stone walls of bureaucracy where Father Brown is concerned. Mallory wears his insecurities on his sleeve, but as a result, he seems paradoxically more inclined to internalize Father Brown's advice than the other two.
"Blind Idiot" Translation: In The Owl of Minerva, a victim's dying words are the nonsensical "Helmet two." Later, Father Brown discovers the victim was a Franco-Brit and his actual last words were "Elle m'a tue" (pronounced ell mah too), meaning "She killed me." Except that the real French translation would be "Elle m'a tué" (pronounced ell mah too-ay). Fluent speakers such as Father Brown and the victim would never make such an elementary mistake.
Harsher in Hindsight: The season one episode, The Man in the Tree comes across as this with the revelation that Sid has spent a year in jail in series 5.
Hilarious in Hindsight: Before he was Father Brown, Mark Williams played Arthur Weasley in the Harry Potter movie series. Alex Price who plays Sid Carter, left the show to be Draco Malfoy in "Harry Potter and the Cursed Child".
Honor Before Reason: While Father Brown is often able to withhold information due to the Seal of the Confessional and often uses it for the greater good, in the episode "The Deadly Seal" he persistently avoids warning Bishop Talbot of a threat on his own life due to the threat being made in the confessional despite the very real chance it could lead to his death. Justified in context: Father Brown is a Catholic priest, and for Catholics, the Seal of the Confessional is absolutely inviolable. The only person capable of granting permission for a priest to break it is the Pope.
Moral Event Horizon: The staff of St. Bridgit's Moral Welfare, featured in "The Bride of Christ", cross it through their stealing babies from their mothers over any moral deformity, and they even had at least some of said mothers locked up in insane asylums, effectively scarring them for seven life terms. Worse, the murders of the week took place because of it—the murderer of the week happened to be one of their victims. It's telling that even Father Brown thought the whole operation was morally reprehensible, well beyond anything he had ever encountered as an amateur sleuth.
Narm: The second death in "The Crackpot of the Empire" would have been a lot scarier if they hadn't used the TIE-fighter noise for the victim's dying scream.
Nightmare Fuel: Some of the murders in the show are creepier and more grisly than others. Special shout out to:
The Shadow of the Scaffold where the viewer sees half-eaten body parts in the pig trough.
The Smallest of Things: Agnes talking and singing in her dead sister's little girl voice when entering into a murderous fugue state, not to mention the requisite creepy dolls in her crime scene dioramas