The first time Spider-Man uses his head was in the second story arc. He attempted to go beat up the Kingpin ended with his wrist crushed and fried by Electro. He did his homework, beat up the Enforcers and Electro, got footage of the Kingpin killing someone, and then leaving in one piece after giving a Reason You Suck Speech speech to the Kingpin. Score one for Spidey!
Even more awesome? The speech mostly consisted of fat jokes.
Kingpin's revenge is also pretty good; after beating up and unmasking Spidey, he reveals that he acquired a small studio that had gotten the rights to Spidey's image from the wrestling company he started out with. That meant that if Spidey attacked him, Kingpin would make money. If Spidey fought crime, Kingpin would make money. If Spidey retired, Kingpin would hire some guy to fight crime in his stead. And make money.
In Ultimate Comics #154, Tony Stark outright tells Peter that when he turns 18, there'll be a job waiting for him at Stark Enterprises.
Everything that happens in the Clone Saga arc. Highlights: Peter telling Fury that he'll do anything he wants if he can have 10 minutes alone with Octavius, and Fury accepting; Mary Jane convincing Fury to let Spider-Man keep his powers by saying Peter looks up to him like a father; the Fantastic Four standing up to a small army of SHIELD equipment and soldiers for Spider-Man.
Bonus points go to the creators simply for the fact that they were able to make a version of the Clone Saga that was actually good.
Three for the price of one: Kong figures out Peter's Spider-Man based off very little evidence - namely that Peter was bitten by that spider at OsCorp, and shortly afterwards Peter became athletic enough to make the basketball team, broke a desk without trying, broke Flash's hand simply by blocking a punch, and Spider-Man turned up. Then when he tries to prove it by kicking Peter in the back, pointing out Spider-Man would be able to dodge it, Peter's Genre Savvy enough to figure out what's going on & lets Kong boot him into his locker, at which point Peter turns around in tears asking why Kong would do that, at which point Gwen Stacy makes her first appearance in the series, pulling a knife & sticking up for Peter.
And then Kong reveals over a hundred issues later that he has, in-fact, always known Peter was Spider-Man but gave up trying to prove it out of respect for Peter's wishes.
After a whole incident with Kitty that got blown way out of hand (due to the aftermath of Ultimatum and mutant paranoia), the Principal of Peter's high school holds a PTA conference to announce his resignation, but lashes out at the parents (who blame him for what happened at the school) for letting their fear get the better of them and tells them to take responsibility for their own actions rather than trying to find someone to blame.
Spidey talking Daredevil out of committing murder.
Peter losing his shit at Osborn for throwing MJ off a bridge. A SHIELD agent relaying information to an at first concerned Nick Fury says:
Agent: He's beating the crap out of him! Fury:(concerned) Who is? Agent: Spider-Man! Spider-Man is beating the crap out of Osborn!
The turning point of Spider-Man's first fight with Venom. In the Ultimate continuity, Eddie Brock is a childhood friend of Peter, so when Brock turns out to be a psychopath and turns himself into Venom, Peter blames himself, and tries to talk Brock down. When Brock sets the symbiote on Peter, he finally decides to stop talking and start kicking ass. The CMOA occurs once Peter grabs a tire and hits the underside of his chin so hard Venom bites through his own tongue. Totally, totally badass.
Peter: Sit down, Eddie!
Perhaps the single most awesome moment in the Ultimate-verse is J. Jonah Jameson discovering Peter Parker is Spider-Man in the "Tainted Love" arc. You'd think he would go and plaster it all over his paper and ruin Spidey's reputation, right? Wrong! He tells his reporters that he knows who Spidey is, but is not going to tell them and will, in fact, use the Bugle to help Peter.
J. Jonah Jameson, one of the proudest and most arrogant Spider-Man characters outside of the villains, actually has character development during the Ultimatum arc. In the midst of the catastrophe, he personally witnesses Spider-Man dive into the flood to save a woman's life, and instead of blowing it off with some rationalization, he actually writes in his newspaper that he had been wrong all along, that he had shirked his duty as a journalist, and delivers one of the most heartwarming endorsements of Spider-Man imaginable.
He also gets a great moment when he apologises to Peter for firing him, opening up about losing his son.
"The Death of Spider-Man" — The Sinister Six (or rather, the Sinister Five), who had to be stopped by The Ultimates last time they teamed up, turn up at the Parkers' home seconds after Johnny & Bobby return from a double date. The two teens don't back down, and when Osborn turns into the Green Goblin, Johnny flames on and takes Osborn out immediately. That's right, the Human Torch freaking one hit kills the Green Goblin. The Torch & Iceman then hold their own against four villains they've never fought before, before Sandman & Electro manage to take them out. At which point a wounded Peter turns up.
Next issue; Peter spends the whole issue taking on the rest of the group, and does a pretty good job despite the bullet wound. They're being watched the whole time by his neighbors who are calling ambulances and the police for him (his mask is off by the way). Eventually the villains manage to get Peter on the ropes, and Electro is about to finish him off when Aunt May comes out of nowhereand shoots Electro, which causes him to practically explode with electric power, taking out the other villains. Then Aunt May rushes down to cradle the still wounded Peter as all the neighbors crowd around. Then Osborn gets back up.
MJ runs Osborn over with a van. When Osborn gets back up Peter then (remember—BULLET WOUND, beating by various supervillains, and just plain exhaustion all around, plus not being quite as strong as the 616 version) picks the van up and squishes Osborn with it. Twice.
Otto Octavius earns one earlier in the arc. When the rest of the Six are all being convinced by Osborn to kill Spider-Man once and for all, Octavius alone basically says, enough is enough, and decides to stop going after Peter for revenge. He realized how little targeting Spidey has actually done for him, and just wants to go back to being a scientist, and work for whoever will have him, deciding that he should just be proud that he had a hand in the creation of Spider-Man. Unfortunately for him, Osborn takes offense to that last part and decides to kill Otto. Which he then does.
In the previews for Ultimate Fallout #2, Aunt May's confronted by a grieving Captain America, who reveals that it was his fault that Peter is dead and that, before Peter took the shot meant for him, he had basically torn down Peter and told him he wasn't ready to be a hero. Aunt May's response? Bitch slap Cap and call out both him and Tony over the fact that it was basically their fault Spider-Man was no more!
Hell, this was enough to get Steve to actually quit being Captain America.
Miles' victory over the Kangaroo, his very first time being Spider-Man, after getting knocked down and kicked around.
Probably the moment that established Miles as a hero was when he decided to jump in a burning building to help the people there, in his street clothes, even though he was afraid of his powers and of getting caught using them.
Miles taking out Electro by beaning him with metal crates before sneaking behind him and gently touching his neck. The fact that Electro is just absolutely creeped out that Spider-Man is back from the dead is fantastic. And to top it all off, he saved SHIELD's asses in the process, when their own efforts backfired due to bad timing or bad luck. This convinces Nick Fury and co. that Miles is a worthy replacement and they decide to give him free reign for a while.
Another one when he beats Omega Red, since it happens so fast. After he realizes he's won he has a little "Hell Yes!" pose in the midst of all the rubble and confused New Yorkers who absolutely do not give a shit.
Also counts as a Funny Moment, but Miles' taking down the Scorpion, despite still being inexperienced and the Scorpion being much larger and stronger than him. It also averts the Running Gag and the uninterested citizens of his last few battles by having them instead cheer for him afterwards.
In the twelfth issue, Miles finally gives the Prowler (his uncle Aaron) what he's got coming to him, saving a busload of innocent people (one of whom was suffocating to death) in the process and either dodging or tanking everything the Prowler throws at him. And the entire time he's telling the Prowler why he's Spider-Man and why he won't let him push him around. It's a good dose of Character Development for Miles, who's been ruled by the fear (of his parents finding out he's Spider-Man, of Uncle Aaron exposing him) for most of the comic.
Miles Morales: You really don't care who you hurt!? What did the world do to you to make you think that it was okay to be this way? That's why I can never do what you do. That's why I can't be like you.
Spider-Men: Although it was more hilarious than awesome, Miles beats 616 Spider-man with the powers Peter does not have.
Issue 13 and 14 of Ultimate Comics: All-New Spider-Man has tons of moments:
In 13, Aunt May and Gwen finally meet Miles Morales in the warehouse where Peter started training as Spider-Man. Instead of lashing out at him, they want to help him. They even don't believe the news that Miles is a killer and give him a benefit of the doubt. Of course, Captain America thinks otherwise.
In 14, Gwen straight up calls Captain America a jerk for not allowing Miles to be Spider-Man. Despite the support of Gwen Stacey, Aunt May, and Mary Jane Watson, Cap, thinking that Miles is still too young and consequently should not be allowed to be Spider-Man, tells Miles that if he continues to be Spider-Man, he'll tell his parents. He leaves and we get this bit.
Mary Jane: What are you going to do?
Miles: I-I don't know. I don't know what I can do. I've been trying to do what I thought Peter would do, but I don't know what Peter would do here.
Aunt May: He'd say: Prove yourself. Prove him wrong.
Aunt May then gives Miles a box and it has Peter's old web-shooters in them. Miles still doubts himself and all three women give him support and a Rousing Speech.
Aunt May: His web-shooters. I thought—I thought long and hard about this—and I thought he would want you to have them.
Gwen: Yeah. Go get 'em, Tiger.
Miles: Wow. But maybe—maybe Captain America is right.
Gwen: He's not.
Mary Jane: But you are awful young.
Aunt May (grabs Miles' shoulders and looks him in the eye): Don't do what we say, don't do what he says, don't do what you think Peter would say...Do what your heart tells you. I've learned a lot in my life and I've learned that life is too short for anything else. Don't do what Peter Parker would do. Do what Miles Morales would do.
In issue 14, Miles gets gored halfway across New York and all that does is delay him for a few minutes while he swings back.
In the same issue, Captain America lands an epic jump kick on the Rhino—both feet right into his face.
Miles hangs his massive brass balls on display by deciding to take down the Rhino using the same method that directly led to his uncle's death—causing his suit to self destruct. With him still inside it. Miles was lucky he didn't become a murderer for real this time, but it was definitely efficient.
The United We Stand arc has Miles joining the Ultimates and saving Captain America from a platoon of War Machines before going toe-to-toe with a Giant Woman and knocking her out.
Meanwhile Miles' dad shoots some terrorist that tried to recruit him. He had a Heroic BSOD but it doesn't lessen the fact that he did what he had to do.
Miles' parents in the Venom Wars arc. If we learn nothing else about them, we know that heroics run in the Morales family. His father, fresh from his above-mentioned altercation with the Hydra terrorists, finds Venom after him before Spider-Man engages. Instead of running or hiding, Jefferson tries to engage Venom with a rock, and is promptly thrown through a car window, putting him in the hospital. When Rio tries to visit her husband Venom shows up at the hospital still looking for Jefferson, where Rio doesn't even think twice before rushing to help people get to safety. Miles shows up, they fight again, and Rio, recognizing who the new Spider-Man is, starts shooting at Venom. Spider-Man eventually beats Venom but it comes at a price for his mom.
Issue #27 has Jessica, Miles, Bombshell, Cloak, and Dagger all going at Taskmaster, who put up a damn good fight.
At one point, Miles gets paralyzed and thrown off a building by Taskmaster. With only one hand functional, he aims a web at the precise spot that lets his momentum swing him down, around, and back up to crash into Taskmaster's back, all without having to move his body, while free-falling, and in the space of like 2 seconds.
Issue #28 has the group finally doling out justice to Roxxon.
Cataclysm in both the Ultimates' Last Stand and Spider-Man have a few moments for Miles and the Power Pack.
Miles Morales, not knowing what to do, hitched a ride on a jet to take on Galactus. Of course, he immediately regrets it as soon as he goes flying. Monica has the jets retreat at the request of Tony Stark, but Miles releases his webbing and lands on Galactus. That is right. A tiny little boy has the balls to land on the devourer of worlds.
Bombshell helped a shopkeeper out after looters with nothing else to lose decide to rob her store. Bombshell, who was a former jewel thief and robber, decided to be the superhero that Peter Parker thought she always should have been. What makes this doubly awesome is that we see Peter Parker's legacy affect not just Miles Morales, but the criminals that he put away.
Dagger, in spite of her previous hatred of mutants, always had the heart of a hero. She always was willing to help people in need no matter the catastrophe, whether she had powers or not. She did not hesitate to be a super hero, and so, when she saw Galactus, she demanded that Cloak teleport them to him so they could fight back. Yes, it was futile, but damn was she awesome to even dare to try.
In the Ultimates' Last Stand #2, Miles offers to go through the interdimensional portal into 616 as he is the only one who has done it before and can be sure to survive the trip.
After a plane has crashed right in the middle of the city, all five teen heroes work together to save the lives of the survivors of the plane (J. Jonah Jameson included among them) as well as any bystanders caught in the chaos, as a fire has engulfed the vicinity. To go into detil, Miles and Jessica gather all the survivors from the plane, while Cloak offers to teleport all the civilians to a hospital far away from the chaos. Farther than he's ever even attempted to teleport, in fact, but he pulls it off. Though the crowd is frightened by Cloak, Dagger takes care of easing the fear of some children and leads them through Cloak and to the hospital, which makes everybody else realize there is nothing to be afraid of and readily accept their help. And to top it off, an explosion sends a large part of the plane's fuselage flying, which is about to crush Miles, Jameson and probably dozens of civilians, when Bombshell appears and blasts the falling piece of fuselage to the point where she disintegrates it, saving everybody's lives.
Peter's response to Silver Sable about how she is gonna get Mary Jane and Aunt May now that she knows he is Spider-Man: "Leave. My family. Alone!"
Sable has managed to get herself out of Venom before he could get a chance to absorb her life force.
Most bosses will do that. Notably, Wolverine slicing his way out.
Most of Spidey's victories over Venom, since Venom could normally tear Peter apart at close range in his early days, but in the game itself Pete seems to be barely out of breath.
The last one in particular stands out. Spider-Man delivers a No-Holds-Barred Beatdown on Eddie, punching the crap out of him. Then, as he walks away, Venom gets back up and prepares to attack him, only for Peter to knock him out with a single punch.