Rio is introduced with a string of these. First, she lures a teacher to his death (thanks to Adaptation Decay, either by falling down and stabbing him when he helps her up, or packing a sloped room with melting dry ice and watching him suffocate at the bottom while she stayed clear of the fumes at the top). Then, she builds a bomb, hides it in a stuffed toy, and brings it up to the room where the Great Detective already expects her to kill his Sidekick—only, instead of building a bomb big enough to kill, it's only enough to seriously injure herself and obscure the distinct physical marking that would identify her as the killer. Finally, when they approach her in the hospital, she's prepared a game: by blowing herself up, she's established herself as reckless and willing to throw her life away. So she tells him that the winner will be the one who can believe in himself, and her companion brings in two glasses of water. She says that one contains an extremely bitter poison called Strychnine, and the other plain sugar; as the two are identical, she doesn't know which is which. She and Ayumu must each drink one, and whoever lives is the winner. Ayumu seemingly identifies the trick, and claims they're both sugar... but the water in his glass is bitter, and it barely touched his tongue. While he's going through his Heroic BSoD, she calmly drains her glass and her companion notes for the audience's convenience that Rio was correct about the winner needing to believe in their abilities: there really was no poison; it was just an extremely bitter cold medicine.
And cue Hiyono the Neutral Female stomping forward and draining the glass herself, proving Ayumu's original theory correct, hitting Rio's weak spot by calling her a coward for lying about being willing to risk her life, and getting Rio to confess to the original murder. On tape. This in turn sets off Ayumu's own Moment of Awesome... but your life has been ruined enough.
The manga has quite a few more Crowning Moments of Awesome, though the two listed above were, indeed, awesome. Kanone has one later in when he lets Hizumi kill him, to accomplish three things: Screw Destiny, something he's been trying to do since he showed up, pull a very impressive Mind Screw on Hizumi, and give Ayumu "hope" (something he's usually in desperate need of).
Hiyono gets some epic moments in the manga, between getting information and blackmailing people, but especially during the Kanone arc. She's supposedly playing the role of decoy against Kanone, who will not kill "normal" people. First, she stalls him long enough for Ayumu to think up a plan, by cutting her own wrist and blocking his way until she collapses from blood loss, forcing Kanone to treat her so she doesn't die. He then sees through every plan, trap, and trick that Ayumu, his sister-in-law, and the Blade Children working together throw at him, until Hiyono shows up with a tranquilizer gun and shoots him, which he technically *did* expect... except that she was wearing a pair of giant cat ears on her head, which threw him completely off-balance. Yes, the one and only battle in which cat ears were the deciding factor. Remember, this is after she had collapsed from blood loss.
Madoka gets hers when she goes all Mama Bear on Kanone for messing with her adorable brother-in-law—for that matter, Kanone is pure awesome in this fight, too. She's a cop, he's a highly trained fighter, they're shooting and throwing knives at each other, and Madoka is doing this in a skirt and high heels.
Ayumu becomes kind of a walking CMOA when he starts tidying up the absolute mess that the series is in the last two volumes, absolutely refusing to break down like he did all the time in the beginning, and even sometimes smiling. (Remember Ayumu is/used to be the worst no-self-esteem brooding angstbucket in the series.) That, or the calm with which he does it all can just creep the hell out of the reader.
The series being loaded with awesome female characters, any occurrence of girls fighting, alongside or against each other, without anyone pulling any punches. This troper particularly cheered at the Ryouko (in a school uniform, using her bare fists) vs Amanae (in an elegant skirt and heels, swinging a friggin' CHAIN) fight in the fifth and last volume of Spiral Alive.
Eyes outsmarting Ryoko in a special game of dodgeball the jocks at Tsukiomi High School play. This is a little convoluted to explain, so it will be done in bullet points. Let's go.
First, the game itself. Player 1 is given three soft, rubbery balls in order to hit Player 2 with. Each ball is in play until it touches the ground. When Eyes asks Ryoko to join forces with him, Ryoko challenges him to this game because it's a test of motor skills, not planning and counter-planning, giving her the advantage. Eyes retorts that it actually is more of a chess game.
Eyes then announces that he won't insult Ryoko's skill by using the most basic strategy— using his first two throws to knock her off-balance and then tag her with the third. He then goes on to announce that he will tag her with the first ball he throws.
Ryoko immediately sets to figuring out his strategy. She concludes that his first throw will be directly into the air above him, and will be intended as a distraction— by announcing that he plans to tag her with the first ball, he forces her to focus on that ball, and can then knock her off balance with the remaining two while she's paying attention to it. Furthermore, he can then catch the first ball, and then use it to tag her, thereby fulfilling his prediction that he will tag her with the first ball.
So Ryoko pays no attention to the first ball that Eyes throws upwards, focusing entirely on him. She taunts him that she saw through his strategy, but Eyes contradicts her: "Checkmate." All of a sudden, the ball tags Ryoko on the back of her shoulder.
Eyes did throw the ball upward, but he threw it at an angle. By announcing he'd tag Ryoko with the first ball, he anticipated that she'd anticipate his strategy, and thus forced her to not pay attention to the ball she thought he expected her to pay attention to. By forcing her to focus on him in order to thwart his supposed strategy, Eyes turned Ryoko from a moving target in a test of motor skills into a fixed target in a test of planning and counter-planning.
Futhermore, Ryoko speculates that he practiced the whole thing. Eyes knows that Ryoko hates him and would refuse to join up with him. He must have guessed that Ryoko would challenge him to a game to get him to back off, so he snooped around ahead of time in order to figure out which game Ryoko would challenge him to (she's undefeated at this), and so worked out a strategy and practiced his parabolic throw beforehand— he's a musician, after all, not an athlete.
Eyes' awesomeness knows no boundaries. There was one time and a Hunter was going to kill him during a concert and Ayumu figured out it was a sniper attack, telling Eyes to delay the show by five minutes while he and Kousuke incapacitate the hunter. What does Eyes do? Start the show two minutes early with a performance involving a lot of candles around him. So the heat rising will distort the hunter's vision from sniping. This also serves as a distraction so Ayumu and Kousuke could attack the hunter easier.