Real Life example. Jim Caveizel, Reese's actor has many friends amongst the Navy SEAL veteran community who help him train for the projects he works on like POI. Reese's Weapon of Choice, the SIG-Sauer P-226R which also happens to be the sidearm of the SEALS could be a subtle nod to them.
How does Fusco know who he can safely call for backup when Elias is breaking into the safe-house? Thanks to his work as Reese's inside man in HR, he knows just about everyone who isn't an honest cop in New York, and called somebody who wasn't on that list.
Extremely subtle but Finch becomes panicked when someone he cares about like a POI or John is around a bomb. Later in "God Mode", we discover why Harold's Only Friend Nathan Ingram was killed by a bomb while Finch had his back ruined in the blast. After this trauma, when someone he cares about is around one, Harold immediately drops everything and if possible is willing to die trying to save them from getting blown up like in the case of John during "Dead Reckoning"
Reese pronounces the Dutch commands he uses with Bear with a heavy German accent. If he was stationed in Germany during his military career, he might have picked up the language there. Then he ends up in Iraq, where he starts working with dogs trained to respond to Dutch commands—and his pronunciation of the Dutch words is colored by his prior experience with German.
The Machine was infected by a virus sent by Decima, but inside that virus was another virus which caused it to protect itself from all attempts to gain control over it. Finch didn't make a virus: he made a vaccine.
In "God Mode" it appears there are a large number of people all in danger on the same day. When one considers that they hadn't previously gotten a number in weeks, it makes sense there would be a backlog.
Why does Root open fire in the asylum three seconds before Hersh steps into the corridor? The Machine is giving her the ability to detect the position of hostiles which was displayed in "God Mode", as a result, she's able to basically lay down suppressing fire and force Hersh to dive into cover, catching him off guard.
Lionel Fusco's days with Stills and the nascent HR hierarchy gave him the skills needed to figure out how to "mock up" crime scenes to look the way a cop wants it to look, so when Carter needs a faked death, Fusco already has everything needed.
"Endgame" reveals how Carter immediately knew Reese had suffered from PTSD back in the "Pilot" episode. Her husband who was a fellow veteran displayed similar traits as their marriage fell apart.
While it is believable that her cover identity could fool Finch, there is no way Root's alias would fool The Machine. It is actually a case of Fridge Heartwarming, as it shows that even with all of the evil things she had done, The Machine still saw the good in Root and considered her worth saving.
Finch landing the plane. Why does he already have controls? Because he probably plays flight simulator games, and he's probably flown virtual jumbo jets, which is why he can make the landing (that, and because he's already a pilot, he knows the basics).
Reese is going to Istanbul. Why? Because Alistair Wesley in "Critical" mentioned they'd crossed paths at the market there, and Reese wants to follow up on that.
Why is Arthur able to remember Finch and all the shenanigans they got up to in college so easily when he has trouble remembering anything else in "Lethe" and "Aleitha"? Think back on the school song they sang together in "Lethe": The last three words are always remember MIT. Furthermore, it's normal among aging people, and even ones with Alzheimer's, to remember long-ago events more clearly than recent ones.
The ISA replaced Shaw with a new operative, Indigo 6 Alpha, who crosses paths with Reese in "4C". While it was pretty humorous that the operative was extremely similar to Reese, it might actually be a clever cover for the ISA. If Indigo 6 Alpha were to get made or happen to operate in New York, they could plausibly pin any crimes or actions he commits on Reese. Given that the only description most law enforcement agencies familiar with Reese is the "man in the suit", it could be easy to confuse them.
For a while, it seemed odd why Finch was insistent with the Machine on not protecting him. At times, it seemed like it was to make sure it did not deviate from its intended purpose to prevent major catastrophes by prioritizing certain numbers (or to not place an "Irrelevant" number over a "Relevant"). Knowing Finch's past seems to spell out a better reason: he is on the run from the government for hacking ARPANET as a teenager. The Machine already showed that it cares about Finch, to the point of matching him with an ideal woman. If his life was in danger, the AI could deviate and send out an Irrelevant number for the government to save, which might end with him arrested. In addition, it might have been a precaution if someone ever managed to gain full access to the Machine, and used it to track him for whatever reason.
Why does Decima need a superfast 89 GHz helium-cooled processor to run Samaritan, while The Machine can run on more conventional (albeit very good) hardware? Arthur had just gotten the Samaritan software to work when the government shut the project down—because The Machine had gone operational. The Machine was at a later stage of development, and its code had been optimized. If Arthur had been given the time to finish developing Samaritan, he would have been able to refine Samaritan's code so that it didn't consume so much processor power.
Alternatively, or in addition, Arthur may just have been not quite as good at inventing AI as Finch. Hence why it took him years longer to reach the same point Finch did. Samaritan is, essentially, making up for a deficit in its design paradigm with excess computing power.
At first, the idea that the Machine would pick Root as its personal agent and avatar seemed indicative that there was something wrong with it. Sure, Root's willing to obey it without question, but she's a sadistic psychopath who has killed an awful lot of people, and the Machine is all about saving people. But "/" makes it clear that the Machine is trying to save her from her own worst instincts - and, in doing so, it's saving all the people she might have otherwise hurt. Not only does it keep her busy helping random people, but it confronts her with a person whose life she ruined and gives her a second chance to save him. The Machine was built to love everybody, and that's exactly what it's doing.
Fusco knows "a place in D.C. that charges by the hour." He's divorced, which suggests that while he might legitimately be bunking in somewhere cheap when in D.C. (hotels are expensive there), it also suggests one possible way his marital infidelities, if any, occurred.
During "A House Divided", why does Reese eventually agree to the seating arrangement during the car trip with Hersh? Because as they don't trust each other, sitting directly behind Hersh puts Reese in a prime position to empty his SIG-Sauer into the man's back in the event of a premature betrayal as they go to rescue their bosses.
For a grass roots domestic terrorist group Vigilance seemed to be particularly well armed and funded compared to most with their members using top of the line European firearms such as Heckler and Koch MP5 submachineguns and having a computer virus capable of knocking out New York's power grid for an extended amount of time. But when John Greer reveals that the Decima Technologies corporation, which specializes in developing such viruses and could easily afford to provide the group with high quality firearms, has been the entity behind Vigilance, it makes a whole lot more sense on how they were able to acquire such equipment.
In "Panopticon," the PoI is Ali, an Egyptian immigrant who owns an electronics shop, and is being forced to build a private wireless network for a drug gang. The network he builds is similar to improvised wireless networks constructed by the demonstrators in the "Arab Spring" protests in Egypt and Tunisia. Explains why they would be going to him for one.
When the characters get their secret identities, it shows just how well The Machine understands them. Finch's job is a fairly obvious fit, and it gives him a position where The Machine can covertly send him necessary information. Reese gets a day job as a cop, because he capable of adapting to work within the bureaucracy. Shaw never could, so she gets a boring ordinary day job and a second job that keeps her from going crazy. Root is capable of playing the chameleon and taking on random changing identities that allow her to take greater risks with discovery.
In Season 4, Shaw has started to take the role of Mission Control more often. Normally you would think a Professional Killer would be totally unsuitable for such technical work until you remember Shaw was an operator for the ISA, an organisation which specializes in communications and surveillance. While she would never be able to hack like Finch and Root do, she would have sufficient experience at coordinating the other members of Team Machine from a desk and other basic technical work.
In "Pretenders", when the Brotherhood goons attack Reese and Walter with their stolen XM 109, the rounds are punching through the van and whatever cover, but are not exploding. Then you realise why, when Shaw handles the weapon: the goons were carrying blue target practice rounds, which still have enough mass and kinetic energy to punch through a van and cinder blocks..
In "If-Then-Else," after teaching the Machine to play chess, Finch discourages the Machine from thinking that the metaphor can apply to real life, insisting that no life is more valuable than any other. This goes a long way in explaining why the Machine would care enough about ordinary victims of everyday violent crime that it would find a way to save them in addition to its mission to prevent terrorism.
In Season 4, as the gang war between Elias and The Brotherhood escalates, Finch is concerned that something is wrong with The Machine because it's failing to predict attacks by Elias. In "Asylum," we find out why: Elias is using New York's long-forgotten pneumatic tube mail system to send written messages to his henchmen. Since the messages that would normally be The Machine's clues aren't being transmitted by electronic means, The Machine can't intercept them.
Someone in Headscratchers asked why Team Machine stored our ol' girl on RAM instead of standard drives, and then I thought about it. Even regular RAM is literally orders of magnitude faster than a standard hard drive or solid state drive, and they were extremely pressed for time.
During the third simulation in "If-Then-Else", the Machine simplifies part of the simulation due to it running low on time. Team Machine's dialogue is noticeably reduced to descriptions of what they would, instead of processing real speech for them. If one pays attention to the screen during the Machine's perspective, the surveillance footage cuts are rendered in lower resolution and the ID boxes are lines instead of dashed boxes.
Out of all of the possible professional killers Root could've hired to take out "Caroline Turing", why hire HR? Other than the show's need to make the finale more dramatic, she likely had her own reasons too: First of all, as they weren't experienced in that sort of criminal field, she probably figured that she had an overall better chance of survival. Second of all, she could easily do what a normal victim would do- namely, want to go to the cops -without worrying that John would actually let her anywhere near them, which would probably put a dent in her plan.
In If-Then-Else, Finch teaches The Machine that the most powerful pieces are not only the most valuble for winning the game, but also for making sacrifices to defeat the enemy. At the end of the season, she sacrifices herself to protect her team. Speaking of, if The Machine is the most valuble and powerful piece, that would make her the Queen - and by extension, also make Finch the King, since he's the least mobile and able to take other "pieces," and taking Finch could be the total defeat of Team Machine. This chess piece parallel could alsp extend to all of Team Machine, making each member a specific piece.
An alternative is that Root is the Queen, as she is able to move most freely, especially in season 4, as she is the only one able to get new identities. She also fallowed the progression that Harold pointed out for the Queen, when she became the most powerful piece by gaining constant God Mode. She also seems to show a willingness to sacrifice herself throughout the season, but Shat beats her to the punch.
In the final episode, some people in the NYPD figure out that Reese is the Man in the Suit and that Fusco was working with him on his vigilante activities. So why is Fusco still on the force at the end of the episode? Samaritan had gotten three cops to try to summarily execute the two of them, and given the whole HR scandal that ran from the end of season 1 to halfway through season 3 - a scandal which Fusco played a key role in exposing - the fact that they tried to murder their prisoners instead of prosecuting them would make it look to most outsiders that they were more dirty cops who forged evidence to justify killing the people who took down their organization. Once that theory hit the news, the NYPD would want to sweep the whole mess centered around a detective that most of New York considered a hero under the rug as quickly as possible.
When Reese is tailing Benton near the beginning of "Cura Te Ipsum", Benton carries a woman's briefcase as they walk to work; this same woman is the lady who recounts being raped by an unnamed co-worker at the survivor meeting, and her photo also shows up in the files that Reese downloaded from Benton's computer. The worst part about this is that although we don't know when he raped her, it happened sometime before the events in the episode and so she has to face the man who raped her at work every day. Not only that, but judging from the fact that he offered to carry her briefcase he makes sure that she has to see him.
Finch was on the phone with Donnelly when Stanton crashed into his SUV with a dump truck. He would have heard the crash, but have no idea what happened. Imagine how terrifying that would be.
The fate of Hannah Frey. She was kidnapped and murdered. But you have to wonder exactly why. If he was just a sick killer, or if it was for perverse reasons... Definitely not helped by the fact that he redid the porch two weeks after she disappeared.
In-universe, Finch tells John he experienced this in "Many Happy Returns": He was originally puzzled why some numbers kept coming up over and over again, and thought it was a glitch... until he realised that these were abused women living with their abusive partners. Fun fact; there's strong evidence that a lot of abuse (up to half of straight relationship abuse, depending on the source) is F>M, and lesbians, depending on which stats you use, have the highest likelihood of all. So how many more numbers is Harold missing?
In "Booked Solid", Hersh notes in his conversation with the Special Counsel that it took him longer than he thought to escape prison. Given that his original goal going in was to kill the four men in suits (Reese and the three mercenaries), the delay might have been him killing off the other two mercenaries. It might have just been to cover loose ends (if they tried to retaliate over Hersh killing one of the trio in "Prisoner's Dilemma"), or in case Reese was a separate threat coincidentally caught with the "man in the suit".
In "MIA", Mrs. Thompson's pacemaker was implanted by Samaritan. Even if she makes it out after Reese leaves her at the end, it's pretty obvious what's going to happen to her once Samaritan's feeds get back online.
Just the simple fact that Leon hasn't been seen or heard from since Season 2's "All In", is kind of unnerving.
Season five makes some of the previous Fridge Horror into Ascended Fridge Horror by revealing what would have happened if the Machine had never existed. In particular, Root remains a remorseless psychopath who ends up working for Samaritan, and the latter has no effective opposition. It also gives us an episode in which multiple attempts are made on the President's life, with no apparent attempts made by Samaritan to stop them. And then a viewer considers that perhaps Samaritan was behind them in the first place.