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.
The short It's the Natural Thing to Do, as well as the later Famous Studios short The Hungry Goat, which feels more like a Tex Avery cartoon with Popeye thrown in as an afterthought.
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.
Ear Worm: "The Man on the Flying Trapeze" gets repeated over and over again in an episode
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 miscalculated amount of iron in spinach, which was 10 times the actual amount. 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.
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.
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.
Willfully Weak: One theory suggests that Popeye doesn't even need spinach to use his full power, and only acts like he needs it to mess with people.
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.