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.
Big Lipped Alligator Moment: In "Morning, Noon and Night Club," Bluto is going around punching out Popeye's face in posters of him and Olive's nightclub routine. He does yet another one after Olive turns down his date offer — and as he walks away, a goat suddenly sticks its head out of the hole and bleats as Bluto tells it off. What a goat was doing behind that wall is never explained, and the goat never appears again in the short.
There's also "Popeye meets William Tell", which for no discernable reason decides to throw our hero into medieval europe and have him encounter William Tell. And it only gets stranger from there.
Broken Base: Some hate the 1960s shorts for relying on limited animation and not having the charisma of the Fleischer and Famous shorts, while others like it for being more faithful to the comic strip.
Crazy Awesome: Popeye given the sheer level of insanity some of his feats.
Fridge Horror: If you know a little about the chemistry of spinach. Spinach and its relatives are high in compounds called oxalates, which when ingested repeatedly over a long enough period of time, precipitate calcium from the blood to yield calcium oxalate, the main component of kidney stones. Ouch. Interestingly, the supposed reason for Popeye's super strength was the amount of iron in spinach. The oxalates would still negate this by binding to the iron to form iron(II) oxalate.
"Funny Aneurysm" Moment: In one cartoon Bluto, in a rare moment of Genre Savvy-ness, destroys the world's spinach supply with an powerful herbicide made from DDT. What it's called in the cartoon, Drop Dead Twice, proved to be appropriate name for the real DDT, at least with regards to birds.
Genius Bonus: From one of the later theatrical shorts, "Insect to Injury" (1956), one gag involves the termites eating Popeye's piano, revealing a harp hidden inside it. Music history fans will take note that the harp was in fact a direct precursor to the piano.
Rooting for the Empire: Some fans' reaction to the racist episode "Big Chief Ugh-Amugh." The Chief, though expressing his desire for a bride earlier through song, didn't actually say anything to Olive about it or make her stay. Instead, he gave her some gifts. She wanted to stay, and it was Popeye who appeared and started picking fights and insulting people.
Seasonal Rot: By the mid-1940s, the Famous Studios Popeye shorts became increasingly formulaic and stale, and the timing and animation took a hit in quality. By the 50's, the series went through such a clear budget crunch that they were forced to make an excessive amount of clip show episodes or remakes of older shorts. Roughly 15% of all Popeye theatrical cartoons from both Fleischer and Famous Studios were either remakes semi-remakes or clip shows, that's roughly 35 cartoons in all! However roughly only 3% (4 total) of the Fleischer cartoons qualify, whereas a whopping 25% (roughly 31) qualify for Famous Studios.
Ugly Cute: Popeye as an infant, of which we get a glimpse in Goonland.
Values Dissonance: The short "I Yam What I Yam" portrays Native Americans in a very racist light. Popeye and Bluto are also frequently quite sexist in their treatment of Olive Oyl, who usually doesn't seem to mind.
Popeye's also beaten up Mexicans, Japanese, African natives, and the like, he's equal opportunity when it comes to brawling.
This would fit with Elzie Segar's original conception of Popeye as a near-indestructible sailor of immense strength long before spinach entered the plotline.
This seems to be all but confirmed for the Fleischer cartoons, where for example in "Can You. Take It" Popeye survives all of Blutos insane "initiation rituals" even being pressed into an iron maiden! And this is BEFORE he eats the spinach!