Base-Breaking Character: Serpentor. There are fans who like him for having an interesting backstory and being at least an inkling more competent than Cobra Commander as well as fans who hate him because he owes his existence to Executive Meddling and see him as something of a Replacement Scrappy for Cobra Commander because of usurping control of Cobra from him.
Broken Base: A lot of fans disliked the DiC Entertainment continuation (though some at least consider the "Operation: Dragonfire" arc decent), but there are also fans who find it a good show on its own even if it doesn't quite live up to the quality of the Sunbow series.
Dork Age: Many people feel this way about the DiC Entertainment continuation, following on the Sunbow series with a smaller budget, Art Shift and a number of other changes that weren't well received.
Senior Cadet Sheila McDermott at the Cobra Academy, a one-episode minor villain. A Dark Action Girl who served as The Dragon to that story's major villain, Commandant McCann, she is popular mainly for providing a name and human face for the regular, usually anonymous Cobra soldiers, who was more of a "realistic" fascist true believer than a megalomaniac supervillain. (Though it probably didn't hurt that hers was an attractive face, either.)
And Raven, the Strato-Viper (Cobra Ace Pilot) from one of the last Sunbow episodes. For much the same reasons, really.
Evil Is Cool: Most Cobra characters, really; even the much-maligned Cobra Commander manages it on occasion. Serpentor might be an exception, while Dr. Mindbender definitely is.
When Scarlett and the Duke android fight in "Synthoid Conspiracy". Imagine if they were real people: She knows he's a fake by then, but it's still a young woman beaten up and almost killed by someone who looks, walks and talks exactly like the man she loves, down to the smallest detail. Of course, she's saved before he does any permanent damage to her, but even so, what does that (realistically) do to her psychologically? Will she ever feel safe with Duke again?
"Glamour Girls," when Cobra kidnap auditioning young model hopefuls to feed to a latter-day Countess Bathory expy. This is obviously pretty bad in itself, but if one ignores the fantastic life-draining aspect, the episode reads very much like Cobra are running a human trafficking ring.
Extensive Enterprises. GI Joe knows this massively corrupt Mega Corp. is run by Cobra, but they apparently can't shut it down. Only two explanations for this are possible. Either Washington is so corrupt in this setting that the government protects a company that finances domestic terrorism. Or, even worse: Cobra is so powerful that they can lean on the government to stay out of their business.
That last bit must be true, anyway. Cobra semi-regularly launches full-blown military attacks on domestic targets in the CONUS, with air support and armor, and the government is by all accounts either unable or unwilling to stop them. So, imagine what a Crapsack World this universe must be: The United States is subject to air raids by an N.G.O. Superpower of fascist terrorists, like Yemen or Pakistan are to American bombings in real life.
The Cobra Academy, which turns out the elite of Cobra's troops. It's run by a dishonorably discharged ex-Marine who is an exemplary soldier and brilliant officer, but who was run out of the Corps because he is also a bully and Sadist who injured, abused and extorted his own classmates as a cadet. Now he's in charge...
Really, a lot of Cobra's cartoonish villainy becomes this if one stops to think about it. For example, having unarmed captives fight wild beasts for entertainment? Within the limits of the show (where, of course, nothing graphic happens) this makes for an exciting scene; taken seriously, it means that the Cobra leaders are depraved sadists on par with the nastier Roman emperors, who enjoy seeing people horribly mauled and eaten alive.
In the animated movie, the reaction from various characters of seeing Cobra Commander after his exposure to a spore pod.
Also in the movie, what all the Joes captured by the "trees" may have gone through, could those trees have had some kind of anesthetic effect, like when a close up of Scarlet later is shown with her eyes closed, or could they, especially Roadblocks detachment from earlier, have been awake the whole time?
In "The Traitor" Duke schemes to sabotage large amounts of Cobra's arsenal by letting them have the Joe team's secret armor formula, which he knows is unstable and thus unreliable. A lot of Cobra equipment is ruined, which is cool and all, but what if Cobra actually found a way to stabilize the armor formula?
Harsher in Hindsight: The idea of a terrorist N.G.O. Superpower with a faceless leader and a legion of brainwashed fanatics taking over whole countries and terrorizing the world isn't as fictional now as in the 1980s. Granted, Cobra is very different from ISIS thematically, but there are some parallels.
Cobra Commander, in the stories where he went into politics. He generally ran a populist campaign against government corruption and incompetence, offering the people simple and drastic solutions to Washington's shenanigans and ranting against the crooked media and the villainy of the Federal Reserve. Like Pat Buchanan, he was a couple of decades ahead of his times...
The episode "The Wrong Stuff" features a parody of The A-Team as one of the television programs used by Cobra to poison the world's minds and ends with the B.A. Baracus stand-in asking to work for the Joes. This becomes hilariously more relevant because the later cartoon G.I. Joe: Renegades had a considerable amount of influence from The A-Team.
Inferred Holocaust: Many of Cobra's attacks are of a type that will have lasting consequences, though these are generally not addressed beyond the end of the episode.
For just one example, the story where they attacked New York and fired heavy artillery into the inner city, blowing down whole skyscrapers. Since nothing indicated it had been evacuated (with panicking crowds running through the streets), there should be thousands dead as an absolute minimum.
"Cobra Stops the World" had them cutting off the US entirely from foreign sources of petroleum, and destroying enough of domestic refining and distribution infrastructure to drive the economy to a standstill. In other words, think 1970s oil crisis on super-steroids. This will totally wreck not only the US, but the global economy for years to come. And depending on just how successful they were—Apparently very much so, since even the military forces were on critical rationing—there might even be starvation and social breakdown in the cities, when food distribution fails for lack of fuel.
Moral Event Horizon: All Cobra leaders dive headfirst through it. Their doomsday plots will generally amount to mass genocide, but they're totally ruthless (and often sadistic) about little things, too. In no particular order:
The Baroness (using mind control) trying to force Scarlett's father to kill her, and what she tries to get Shipwreck's nephew to do later.
And with Zartan, the way he treats Zarana. See Tear Jerker, below.
Averted on the personal level with Storm Shadow, who is not sadistic and can be a sort of Noble Demon on his better days. However, he's still working for genocidal fascists. In one story, he even tried to destroy his own country, Japan, for them.
If they hadn't already qualified separately, Cobra Commander, Destro and the Baroness cross it jointly when they laugh at a young woman about to be eaten by a polar bear.
Some Anvils Need to Be Dropped: While "The Greatest Evil" is yet another Drugs Are Bad episode, it does demonstrate a fairly realistic representation of the consequences of drug addiction by showing Falcon's addiction to drive a wedge between him and Duke, a Cobra member's sister ending up in a coma, and the Headman ultimately meeting his end from overdosing on his own drug.
Take That, Scrappy!: Serpentor's transformation into an iguana in the DiC series' "Operation: Dragonfire" arc can be seen as a satisfying comeuppance by those who see him as a Replacement Scrappy for Cobra Commander.