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These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
His behavior throughout the series is that of someone completely detached from the real world—with Ted even having to draw him a chart describing the differences between fantasy and reality—and moreover Dougal's thoughts and words do occasionally conform to reality and to varying degrees. In addition, many of his behaviors throughout the show are not outright stupid but just plain weird and nonsensical. He probably only became mentally ill sometime in his adult life, after becoming priest.
Broken Base: More like "broken periphery demographic." Catholic priests either think it is hilarious and harmless satire or insulting and poorly researched slander.
The Rugged Island priests' song "The Miracle Is Mine" from the same episode qualifies too. It is hammy and overblown (in the vein of many Eurovision entries), it has a vague religious lyric to it, and is backed up by a full choir and orchestra. This is a good Establishing Character Moment for Dick and Cyril, because they are shown to have a genuine musical talent and planning, whereas Ted and Dougal do not. However, due to the biased voting (again, which happens in Eurovision in real life), Ted and Dougal manage to win anyway.
Ending Fatigue: The Christmas special. Even the show's creators complain about it in the DVD commentary.
To a lesser extent the two part episode "Escape From Victory" and "Kicking Bishop Brennan Up The Arse".
Ensemble Darkhorse: Bishop Brennan, who gained legendary status after appearing in a total of three episodes ("The Passion Of Tibulus", "The Plague" and "Kicking Bishop Brennan Up The Arse".
Fountain of Memes: Even a decade and a half since the last episode aired, the show is still highly quoted in Ireland and many current events will have a Father Ted meme attached to it.
Fridge Logic: How on God's green earth did Dougal become a priest? Frequently asked In-Universe, especially by Ted. ("Did you win a "collect twelve crisp packets and become a priest" competition?") Of course, given some of the other priests out there in universe, Dougal is hardly the worst of them.
The flashbacks of Father Jack in his days teaching in a Parochial School during the first series episode 'Grant Unto Him Eternal Rest' definitely qualify as a "Funny Aneurysm" Moment given the subsequent scandals in Ireland regarding exactly this sort of abuse in Irish Catholic Schools and Orphanages.
One of the Bishops dying of a massive heart attack as lead Actor Dermot Morgan would later die from the same thing.
Even worse, Ted grabbing his chest in shock when Mrs. Doyle surprises him with a cup of tea when he turns on the light.
And the three 'Death' Tarot cards he chooses from the fortune teller's deck in the first episode.
Averted in the finale; the writers originally intended the series to end with Ted commiting suicide, but this was replaced with a clip montage when Dermot Morgan died.
Another cringeworthy one is Jack's P.E teaching.
Teaching Nun: He's also asked me to remind you that it is very hot today, so there will be no need for your tracksuit tops.
The dancing priest dying from having a heart attack due to too much dancing. As seen in the Harsher in Hindsight example, Morgan's constant reshoots of his dance scene in "Going To America" exacerbated the heart problems he was feeling during filming, and prevented him getting treatment in time. So effectively, what was one of the most absurd and improbable deaths in Father Ted ended up coming true.
The titling of the penultimate episode "Night Of The Nearly Dead". Dermot Morgan died after the filming of the following (and last) episode "Going To America".
Harsher in Hindsight: Dermot Morgan was actually having heart problems during the filming of the Shaft scene of the last episode "Going To America". However, Tommy Tiernan (who played the suicidal priest) kept flubbing his lines, and the producers would not let Dermot go until it was finished. To make matters worse, Dermot had to do dance moves in this scene, and doing them repeatedly exacerbated his condition. He died less than 24 hours later. If the reshoots of the scene had been stopped and Dermot had gotten treatment, there is a chance he might have lived. However, there is equally a chance that he might have died whilst away and the rest of the episode would never have been filmed.
The set-up is that the Catholic church deals with miscreant priests by quietly shuffling them off to some out-of-the-way parish until it all blows over. Sadly, this turned out to be the case in real life too, and with worse offenders than Ted, Dougal and Jack.
Mexicans Love Speedy Gonzales: As noted above, it's incredibly popular in Ireland. Yes, it was created by Irish writers, had an Irish cast, and was set in Ireland, but it was still intended for British audiences, making its popularity in Ireland an unintentional (or least secondarily intentional) byproduct.