Action Dad: Minamoto: marksman, scientist, homemaker. In one episode, he forces his kidnapper to eat her vegetables and take a bath, and washes her panties and then uses the opportunity to his advantage (No, not like that). He cleans in his spare time, and his bento lunches are worth fighting a war over. He even has a moe moe apron.
All Bikers Are Hells Angels: Subverted. Turns out that the bosozoku gang that Naomi investigates are loud and unpolished, but really a bunch of softies, and when a crook blackens their name, they don't take it at all well.
All of the Other Reindeer: A very pleasant aversion comes when the Children rescue some bank employees from the Normal People. The bank manager thanks them and says that he considers the Normal People to be idiots.
Angst: Minamoto spends many a sleepless night worried about the legal and social discrimination faced by espers, especially the three under his care. It's kept far, far away from wangst because it isn't self-centered and he generally focuses that energy on productive endeavors, such as giving the Children the opportunity to go to school.
Anime Hair: Most of the cast is usually in uniform, and the same age: you really need the hairstyles to tell them apart in combat scenes.
Anti-Villain: Kyosuke. He's a marginally sane, sociopathic terrorist who's preparing for a war against Muggles. He's also helping, protecting, and providing a home for espers who have nowhere else to go because the Muggles rejected them, doing his level best to help Minamoto keep Kaoru alive and sane (even if it's for his own purposes), and he's about the only one who's doing anything to stop Black Phantom before Kaoru gets her Messiah powers.
Becoming the Mask: Subverted, Feather (now a separate entity) reveals that Yuri, Phantom Daughter's "civilian" personality (disregarded by her Emotionless Girl and Ax-Crazy personalities as a mere "puppet") is how she would have been is she hadn't been found by Black Phantom.
As the manga progresses, this gets more extreme - In the beginning, Shiho would just tell someone's emberassing secrets. Recently ,her first proposal to deal with a bad guy hidden in a cloud who had pissed her off was using thermobaric bombardment, and her second was using a nuke
With the possible exception of Shiho, who calls BABEL's new submarine a "victory for CG" when the girls are talking about how cool it is.
Then, there's the OVA. The robot duplicates of The Children explain to the audience (but confuse the rest of the class) about the making of the OVA and the possibility of a season two.
Bridge Bunnies: Hotaru and Natsuko, the receptionists at BABEL, who later (after being promoted to field agents) get replaced by Chizuto and Tae.
Brutal Honesty: Shiho's mother adopts this as a coping mechanism for the mind-reading aspects of Shiho's powers.
Bullying a Dragon: Done to all espers as well as some normals. Considering what espers can do and how it's the normals' fault some of them go bad in the first place, a lot of them may just be Too Dumb to Live.
In episode 40, the Karen Girls, the ones who sang the OP, make an appearance.
In episode 27, Kaoru blames her behavior on "that Nikkumaru": Nikkumaru Sarutobi, the skirt-flipping ninja hero of 80's manga/anime Sasuga No Sarutobi. The Karen Girls also appear on the magazine Hyobu is reading, along with Ghost Sweeper Mikami.
Canon Immigrant: Andy Hinomiya and Yugiri were characters originally conceived for the "non-canon" series The Unlimited, believed to just appear in the manga's omakes, until they finally appear in the main story.
Shiho: "The pain and hate of the fish which were brutally murdered... it's delicious."
When she first met Aoi, she bullied her and mutilated her toys because she figured people were going to be afraid of her anyway, so they may as well hate her first (she also did it because Aoi annoyed her by being a normal kid worried about her relatively non-destructive power). This was when she was four.
CreepyHalf-Identical Twins: The Little Mice. In their omake introductions, the boy disguises himself as his sister to be with her due to the nature of their powers, and both greet their "sempai" Naomi with "I'm going to use you as a stepping stone."
How about his introductory storyline in the manga? Where his response to being shot and slowly bleeding to death is to calmly use his powers to stem the bleeding and keep the bullet from entering his heart, then directing several others in its removal. All the while, perfectly calm, despite the obviously excruciating pain he's in.
And now, in a recent arc, he decks himself out in warpaint and CHOPS THE HEAD OFF OF A LION.
Dark Messiah: Kaoru, as the Queen of Catastrophe, is prophesied to lead a war against non-espers.
Date Peepers: The Children spy on a supposed date between Naomi and her art teacher.
The chief get his date peep too... using spy satellites.
Kaoru spends an afternoon jealously following Aoi and Shiho around on their dates. She eventually teams up with the Chief to take date peeping past eleven.
A particularly hilarious variation happens when Minamoto goes to his home town for a marriage interview with a girl which happens to be his childhood friend. Since the Chief and Fujiko and Kyousuke are interested in this, and bring their men with them to participate in peeping, it ends up in a chaotic all-out battle between BABEL and PANDRA.
Evidence 1: In an occasion he creates a situation for Minamoto to ask for his help, but he realizes it is a trick to make him his mindless puppet
Evidence 2: He tempts Minamoto once more by having him spend a day with The Children as his 12-year-old self and offers him a chance to make it permanent
Evidence 3: The president of Comerica promises his younger son, a humongous brat and level-7 esper, to PANDRA if Minamoto can't get him under control (just one all-powerful demon child with issues? no problem!).
What do his crotch blasts do? They make things "harder" or "softer".
A non-sexual example of this trope is the scale used to describe the power levels of espers. The Description of the effects of espers at various levels appear to be taken almost word for word from the JMA earthquake scale, making the heroines literally Persons of Mass Destruction.
A pretty unnerving use of it, considering her age.
Earn Your Happy Ending: Minamoto, after 50+ episodes of constant devotion has advanced the chance of a good future by a grand total of 2%. Even so, Fujiko treats it as a significant accomplishment, and The Children tell him that they're aware of, and grateful for everything he's done.
Other thing is they seem sometimes to even like each other.
Even Evil Has Standards: ESPer supremacist group PANDRA is totally opposed to the brainwashing esper Terrorists Without a Cause Black Phantom. Black Phantom is slightly disturbed at themselves for turning children into Tyke Bombs, and remedies this by "not letting them get infected with emotion so they can live unaware they they're human".
Everybody Lives: In the anime, there has not been one individual killed yet, despite how narrowly several people have escaped death. Lieutenant Ikyuugo is the most blatant example, since this leads to a Like You Would Really Do It effect that slightly trims back the drama.
Where by Deus ex Machina you mean the plot point the previous 36 episodes have been building up to, but yes.
Hyobu was trying to save his opponent, not kill him. He was about to give up and take off the gloves when Kaoru smacked him into a wall. A minion later notes that most of his injuries came from Kaoru.
The assassin did manage to give Hyobu trouble thanks to his ability to manipulate bullets. Specifically, he manipulated the bullets that had been left inside Hyobu's body for the last few decades after his former army superior tried to kill him.
Expy: Most character will seem very familiar if you've read Ghost Sweeper Mikami (Shiho, for example, seems to be growing up with all the morality and softness of Reiko), but the only true expy so far seems to be Dr. Chaos / Col. Grisham.
Taken to a whole new level when Kyosuke hijacks a country to allow Mio and the other young members of PANDRA to attend school with The Children under the aegis of diplomatic immunity.
Furo Scene: In a twist, Minamoto gets one of these when the children teleport at home while he's taking a shower. He even chases them off throwing things.
Fictional Counterpart: Except for Japan, all other countries have fictional names, slightly different from their real counterparts, including Comerica, Chena, Jermany, Idaly, Ingland, Franse, Paland, Robiet and Vrajil.
Gagging on Your Words: In an omake Kyosuke is forced to admit that he and PANDRA can't remove the bomb implanted in Sai's head, but he would rather die than say Sakaki is better than him at something.
Gambit Pileup: Fujiko and Kyosuke both have their own plans for how to Screw Destiny. Minamoto has his own ideas and doesn't want either of them telling him what to do. We still don't know Black Phantom's true objectives (if they have any).
Gambit Roulette: Usually, Minamoto's planning is completely believable, but in the episode where he gets the Normal People to threaten Kaoru and fall into his trap, even if we accept that he had concealed an emergency ECCM and limiter release program in his laptop for just such a contingency, which admittedly is in character for him, he still got lucky in that Kaoru was the Child being threatened instead of Shiho (who lacks direct combat powers).
Go-Karting with Bowser: The Children and PANDRA's younger members go to school together, and often spend their free time with each other. Kyosuke and Kaoru also frequently hang out together, even going on dates from time to time.
Hates Being Touched: Inverted trope. People go out of their way to avoid physical contact with psychometers. Sakaki expresses open jealousy that Shiho has people willing to hug her and hold her hand.
Heart Is an Awesome Power: Psychometry. It seems like the kind of thing that wouldn't be much use in combat (and it actually isn't at first), until Shiho explains that psychometry grants Improbable Aiming Skills and lets her use any weapon like a prothe second she picks it up. Don't believe it? Toss her a shamsir and see how long you live.
Hollywood Density: Kaoru stuffs her shirt with gold bars without ripping it, even though she has no powers due to ECMs.
Honor Before Reason: The Children get very agitated when Minamoto does the pragmatic thing rather than the right thing. In one instance, he lets the Big Bad get away rather then subject The Children to the sight of a normal human shooting an esper in cold blood. Unusually for the this trope, it works for the best. The fact that Minamoto honestly tries to live up to this standard has Kaoru reacting to Kyosuke's nastier actions and beliefs with horror and disgust in later episodes.
Kaoru goes way, way out on a limb to trust Mio and the other PANDRA kids at their word, even though she is very much aware of the risk.
Hot-Blooded: Kaoru, so very much. Fujiko was this when she was younger; nowadays she takes her craziness in a different direction.
Improbable Aiming Skills: In the manga, Sakaki manages to hit a power staff carried by a man standing on top of a passenger jet in flight, using a handgun. It's implied that psychometers like Sakaki and Shiho use their powers to aim.
He can also throw scalpels down the barrel of a helicopter-mounted gun, apparently.
In the Name of the Moon: Complete with sentai-style smoke-bombs at the end. Lampshaded by Aoi, Minamoto, hell anyone who's watching as they all think it's a bit silly. The speech is never the same one twice. Even when Aoi's going on a mission alone, she still does a heroic speech. While it doesn't turn out completely perfect, the finale even has The Hound trying their hand at one.
Invocation: Sure, Minamoto doesn't need to call out "Released!" when he deactivates the limiters with his phone, but it looks cool.
Ironic Echo: In an effort to kill The Children, the Normal People are willing to let three civilians suffocate to death in a vault, and say that it is "a small sacrifice". Much later, some members of PANDRA blows up a plane to flush out a Black Phantom agent. They call it "a small sacrifice".
Jekyll & Hyde: Arabian princess Sera and her colder ancestor Masara.
Kick the Dog: Just in case there was any worry of the PANDRA sliding into Anti-Villain territory, they blow up a plane to smoke out a Black Phantom agent, smugly explaining that normals have no right to live. In front of Kaoru. Given that Kyosuke has bent over backwards to paint PANDRA as La Résistance, you'd think he'd have explained the whole "hearts and minds" thing.
It's then followed up by Kyosuke erasing Kaoru's memory of the incident so that she'll be more likely to join them in the future. Not to mention the whole "trying to turn a young girl into a living WMD against Normals despite knowing that this will eventually lead to her death."
Kill Me Now or Forever Stay Your Hand: Kyosuke does this to Minamoto when the latter has the blaster drawn on him. He actively taunts Minamoto into shooting, because he knows that if Minamoto does take him down, Kaoru will come away with a grudge and the esper-human war will be assured.
Mood-Swinger: Naomi, though not without reason. Her supervisor is something of a Dirty Old Man. Her tendency to quickly go from calm and collected to throwing Tanizaki into a wall made Naomi's Code Name change from "Kitty Cat" to "Wild Cat" early in the series.
Mood Whiplash: The series frequently switches tone from serious to comedic, sometimes within a few minutes (or, in the manga, in the space of a few panels). Even Minatmoto's vision of him killing Kaoru becomes an example when the characters themselves start parodying it.
Multiple Demographic Appeal: Despite the girls supposedly being the stars of the story, Minamoto, Sakaki and Hyoubu's popularity usually give them a run for their money, to the pleasure of the female readers (fujoshi or not). This gets more evident in the spin-off series with all three getting even more screentime than them.
May be subverted as those were power-nullifying cuffs they put on him.
That and he receved the mother of all nutshots, which was jus' about where Okama was firin' his beams...god that didn't sound right at all...
No Social Skills: Hatsune. Without Akira's help she'd probably be roaming the woods somewhere.
Not Quite Flight: Aoi's Chain Teleport. It's a rather bumpy ride, but it gets you where you're going at supersonic speed.
Occidental Otaku: Ex-Black Phantom Espers are the most Otaku like of the cast (the two boys are always going on about Moe and 2D, while the girl is a Yaoi Fangirl), but it is heavily hinted that Black Phantom gets its espers from outside Japan in most, if not all, cases. Also one of the Comerica agents is a massive fan of Japanese Culture, though Traditional culture more than Otaku culture.
It is implied in the manga's Chapter 203 and Episode 46 of the anime that the otaku behavior of Bullet, Tim, and Patty is an odd sort of coping mechanism for their loss of memories after getting freed from Black Phantom's control.
The supplement 4koma for Chapter 198 show that Yuuri (schoolgirl form) has inexplicably collected Chil Vermillion anime figures without Mirage's approval, and the supplement 4koma of Chapter 234 shows that Black Hanzo wearing a "Maruto" shirt - which prompts Mirage to ask "are all of our Espers into that kind of thing...!!?"
Older Than They Look; Kyousuke doesn't look a day over 20, but he's really in his 80s. The same for Fujiko.
Omake: Plenty of them. In a slight subversion of the trope, some of them are more or less Canon.
Omniscient Morality License: Fujiko and Hyobu both claim this because they know one of Lieutenant Ikyuugo's premonitions; Hyobu believes it's inevitable and must be brought about, Fujiko has a plan that might stop it. Minamoto doesn't buy either.
Out-Gambitted: In the more light hearted stories, Kyosuke frequently assures his comrades (mostly Magi) that he has a brillant plan to lure Kaoru to PANDRA. Hilarity Ensues.
Overnight Age-Up: Minamoto is hypnotized by the Big Bad into seeing the girls as fully-grown women. The girls decide to have fun with this much to his dismay...
Person of Mass Destruction: Pretty much all characters treat those three girls like that in the beginning. Minamoto might be the first to think of theml as actual people. Kaoru definitely is one and is still growing into her powers; Fujiko and Hyobu are already there. Aoi and Shiho don't cause big blasts, but Aoi can teleport people into solid matter, among other things, and Shiho's power suddenly becomes really scary when applied to international politics or stuff like launch codes.
Power Incontinence: When Kaoru suffers a head whack early on, her psychokinesis goes out of control. There's a reason for the Power Limiter. Other espers are later shown to have had trouble with their powers while young.
Actually, his powers are mentioned in same chapter he's introduced; he's a hybrid type with use of 7 power types between 2rd and 4th level, however as he can combine their effects it does make him far more powerful then some one with only 1-2 of those powers would be. Not to mention 80-ish years of practice
Power Limiter: Very essential here. High level espers without limiters can unintentionally damage themselves and others.
Power Nullifier: The ECMs are the wide-area weapon version of the above limiters. And even then they're no match for Hyoubu.
BABEL has the ECCM, which nullifies the nullifier.
In the pilot chapters of the manga, Minamoto did actually have (unknowingly) an esper ability - he was a living ECCM.
Power Gives You Wings: Kaoru manifests a pair when facing off against one of Mio's copies merged with a bulldozer. The second opening theme (titled "MY WINGS") plays this up even further by granting all three of The Children wings.
Prehensile Hair: Appears to happen to Chisato's hair extensions in an omake. It's really just Kaoru's psychokinesis.
Promotion to Parent: Minamoto has his current job is because he's the parent figure Kaoru and the others need but never had. In somes cases he's more responsible than their actual parents.
Punch Clock Villain: Kazura invokes this claiming PANDRA and BABEL should get along because they work in the same place. Subverted when Aoi retorts that cops and robbers work in the same place. Double subverted when Aoi and Kazura proceed to get along just famously.
Screw the Rules, I Have Supernatural Powers!: A running theme. PANDRA believes that they don't have to obey the rules because they're powerful enough to blast their way through anyone trying to stop them (and in Kyosuke's case, he's pretty much right), and the threat of this (along with plain old bigotry) is the impetus behind the Normal People's terrorism. BABEL is caught in the middle, and it doesn't help that not all of their members are perfectly responsible with their powers either. Many one-off villains also have this as their motivation, such as the telekinetic bank robber.
Stay in the Kitchen: Minamoto is a credible leader, fighter, and tactician, but he's on a team full of the world's most powerful espers, often fighting another team of the world's most powerful espers, and his teammates occassionally tell him to get out of combat. Kyosuke loves rubbing Minamoto's face in this.
The Strategist: Minamoto's second-biggest contribution to the Children (after being the Team Mom) is through cunning ploys that would make Zhuge Liang proud.
Strawman Political: Subverted with the radical normals. While a lot of the anti-esper stuff they do is over the top, it does get taken to that extent with real life hate.
Superpower Lottery: Level 7 espers in general are vastly more capable than espers at lower levels, even Level 6s. Kyosuke, who is basically a Reality Warper thanks to his phenomenally precise telekinesis, is broken even by Level 7 standards..
Teen Genius: Minamoto, in his past and when he was hypnotized and had his memories regressed to when he was 12.
Theme Tune Cameo: The Children sing an off-key version of the first ending theme in a karaoke bar at one point.
Time Skip: In the last episode of the anime (and a similar point in the manga), the show jumps ahead a few years, the most prominent change being that The Children are a little older and have spiffy new uniforms.
Talking to Himself: Episode 24 has interaction between Momotaro and Mio, both voiced by Rie Kugimiya.
Theme Music Power-Up: "Over the Future" is played at least once an episode during a battle or similar situation.
The Revolution Will Not Be Civilized: At least three different groups. Exaggerated to the point of parody when one of the Normal People goes "We're the Normal People! We'll make mistakes that are careless and unpleasant!"
Wham Episode: OVA episode #1 sends the message loud and clear in the first two minutes: the OVA isn't going to be nearly as lighthearted as the TV series.
The "As Time Goes By Arc" in the manga. HOLY crap...
What Measure Is a Non-Human?: At first, Minamoto thinks that Momotaru is a dangerous animal to be put down rather than a being with rights. Kaoru doesn't take it well. Later in that episode, this is discussed when Kyosuke comments that he doesn't see himself as human and humans don't either, and he challenges Minamoto to shoot him and prove his point. Of course, the Normal People see all espers as nonhumans who should be destroyed.
What Measure Is a Non-Super?: The flip side of the above. As far as he's concerned, Kyosuke and other espers aren't part of humanity, and humans need to be destroyed for the protection of the esper race. Minamoto and BABEL try to Take a Third Option and work for harmony between humans and espers, while The Children don't even make us-and-them distinctions in the first place.
Wife Husbandry: Attempted, but only partially successful: Naomi's supervisor Tanizaki engaging in this is the reason she changed from a sweet and quiet girl to a Mood-Swinger once The Children helped her realize it.
Later, when Minamoto and Aoi visit Aoi's home during a mission, her father and brother worry that Minamoto might be trying something similar with Aoi. Aoi's obvious fondness for Minamoto does not help matters.
Wise Beyond Their Years: Although ten, Shiho has a pretty good idea of how dark the world can be thanks to her powers. Minamoto finds it a little disconcerting that even solving murder cases by psychicly extracting clues from the murder weapons doesn't really upset her.
It's not that it doesn't upset her, she gets stressed out when she doesn't get enough crazy murders to solve.
10-Minute Retirement: Minamoto finally blows his stack after catching Fujiko and the Children playing games with him during a mission. He comes back after Fujiko spends an episode messing with his head and he blows his stack at her.