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.
Awesome Ego: Creed is one of the best boxers in the world and knows it, his fights are always given hype like you wouldn't believe (see Big Lipped Alligator Moment). His ego winds up biting him in the ass twice though; first when he plays around with Rocky at the start of the first movie instead of really preparing, and secondly in Rocky IV when he challenges Drago in spite of being much older and assuming Drago was less skilled than he turned out to be.
Big Lipped Alligator Moment: Arguably, Apollo Creed's entrance in his fight against Drago. It starts with an over the top performance that belongs more to a Broadway musical than a boxing ring, has James Brown singing, and has Apollo dressed up as Uncle Sam and dancing. Drago, who is usually The Stoic, is utterly bewildered by this spectacle, and probably qualifies as the Only Sane Man for this scene.
Paulie's robot buddy in Rocky IV. Especially Apollo's reaction to it.
Ensemble Dark Horse: Apollo Creed's nameless trainer in the first film gets an expanded role in all the subsequent ones, including getting a name (Duke). He ends up being the only character/actor other than Stallone and Burt Young to appear in all six films.
Esoteric Happy Ending: Sure Mason won his match in Rocky Balboa but don't tell me the media wouldn't have a field day by poking alot of holes at his victory. The main one being that Mason couldn't take down a guy that haven't been boxing in decades.
Fridge Brilliance: Rocky Balboa works as either a direct sequel to V, a sequel to just Rocky that ignores the four other films, or as a standalone film. One does not need to have any direct knowledge of the prior Rocky films to enjoy it. How awesome is that?
Though he was a former champion in Balboa, so perhaps the third in a trilogy consisting of Rocky, Rocky II, and Rocky Balboa.
Him being a former champion works just as easily as Rocky's backstory. Still no prior knowledge necessary, which is pretty snazzy.
"Funny Aneurysm" Moment: In the MAD spoof of Rocky IV, after Apollo Creed dies, Adrian mentions being nervous because Mickey died in III, and now Apollo, meaning she's next if there's a V. While she doesn't die in V, she does die between then and Balboa.
He Really Can Act: A lot of younger viewers who only know Stallone from action movies, where he (competently) plays very similar archetypes, are rather surprised at his brilliant performance in Rocky.
Also, Mr. T shows some surprisingly understated moments as Clubber Lang in Rocky III, particularly when he quietly makes his "prediction": "Pain...."
Hilarious in Hindsight: In Rocky IV, the depiction of the new (at the time) Soviet premiere Mikhail Gorbachev. Probably one of the nicest leaders in Russian history!
It's hard to call Gorbachev "nice" : he was a pure product of the Soviet Communist Party elite, most of his mentors and people who made him Secretary General were leaders of the KGB and the Soviet Army, he was a Magnificent Bastard who efficiently and ruthlessly eliminated (politically, not physically) many of his opponents in the Politburo and his real primary goal was pure Soviet national interest : preserve the USSR's status as a superpower through intense reforms (not that any other leader doesn't do the same). However, it is true that he quickly tried to appear as a nice and wise leader, looking only for peace and such. And this worked extremely well in America and elsewhere in the West (the infamous "Gorbymania"), making his depiction in this movie hilariously stand out.
Not so much for the film itself, but Weird Al's "Eye of the Tiger" parody "Rye or the Kaiser", about a former boxing champ opening a restaurant becomes this when you watch Rocky Balboa.
In the first movie, Rocky walks lil' Marie home. What was one thing he asks her? "Do you have a boyfriend?" Granted, it was in the context of him encouraging her to stick with only good people, but it's funny when you see how Marie and Rocky interact in Rocky Balboa, including her pecking a kiss on the lips.
Although, they don't exactly Get together. Marie, after all, acknowledges Rocky's wife on several occasions, and he appreciates her respecting Adrian.
After completing production for Rocky II, Sly said in an interview "But there'll never be a "Rocky IV." You gotta call a halt.".
Ho Yay/Foe Yay: Apollo and Rocky. The third film's training montage includes them hugging while splashing through the ocean, and Apollo lends Rocky his old shorts.
Memetic Mutation: Any big, strong, Russian character with a mean streak (or just a flat-out desire to win) will have Ivan Drago's famous line "I must break you" tied to them at one point or another.
The concept of the movie itself. Calling anything a Rocky moment, a Rocky story, or what have you, is pretty much synonymous with an underdog story. Rocky is a Trope Codifier for the underdog.
Moral Event Horizon: A minor one for Tommy Gunn in Rocky V. If clearly betraying Rocky wasn't bad enough, he goes out of his way to be a total jerkass to him at a bar, and even punches Paulie out just because he told him to leave. That was enough for Rocky to want to beat his butt.
Reality Subtext: The unexpected success of the first movie, which made a star out of Stallone, mirrors the actual plot of the movie, and subsequently winning the Best Oscar.
Rocky II as well. After the unexpected success of the original movie Sylvester Stallone was propelled into stardom, but his follow up movies were both critical and commercial failures. He seemed destined to be a one-hit-wonder. Only when the decision was made to make a sequel (there was not one originally planned) and it too became a success was Stallone on the A list to stay. Rocky II and to a certain degree Rocky III follow a similar story line.
Sequelitis: The franchise was widely accused of this, becoming a frequent source of mockery as the series progressed. Averted by Rocky Balboa.
So Bad, It's Good: Rocky IV is often considered as this (in contrast with V, which is just considered bad), what with the over-the-topness, good ol' Cold War patriotism, as well as the antagonist's one-dimensional character and Rocky's narmy speech at the end.
Strawman Has a Point: Rocky IV. Ivan Drago rebels against his Russian handlers and yells that he isn't boxing for Russia, but for himself. Somehow, this makes him a selfish jerk instead of a wordless tool suddenly desiring freedom of choice. Of course, he earlier accidentally killedmurdered a certain former champ, so he can't really gain any sympathy.
Apollo was arguably responsible for his own fate as much as Drago was, given he demanded that Rocky not thrown in the towel for him.
Not to mention the way the movie treats holding the match in Russia. Death threats being made against Drago was treated as an excuse. In real life, it would have been a very valid reason for him to never return to the US.
The death threats are probably treated as a joke because Drago is enormous and surrounded by security. That said, it should be obvious that all the muscles in the world won't save you from a psycho with a gun, especially if you're in hostile territory during the middle of the Cold War.