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.
Between Neal and Sara the insurance bounty hunter. It blossomed into a make out session during a blackout, then blossomed further into intense awkwardness.
Between Caffrey and pretty much all of the female criminals. Some of the male ones too.
Between Neal and Keller. More so on Keller's side than Neal's, since whenever they meet up Keller always reminds Neal of the "good old days" and tries to convince Neal to go back to them.
Between Diana and Abigail which makes it difficult since Abigail's an active crook and Diana's undercover in order to catch her. Plus, Diana broke up with Christy so the attraction comes at a poor time.
Hilarious in Hindsight: In the seventh episode of the first season, they catch a lackey because, upon exiting a secret tunnel, he turns to look at a girl and gets caught on camera. We find out the meaning of the following exchange a few seasons later when we learn exactly how Peter caught Neal the first time. The affected casualness of the last two lines really sells it. And it even works at the time because Neal always, always hits on every woman in sight. Successfully.
Peter: Guy steals three point two million in diamonds and we get him because he can't resist a pretty face. Neal: Happens to the best of us. Peter:[pointed glance] Neal: What? Peter: ... You know what. Neal: No I don't. Peter: Yes you do.
Magnificent Bastard: Neal, Peter & Elizabeth. Curtis Hagen even calls Neal a 'particular kind of bastard' in the pilot.
Mozzie. And part of it is the skill with which he makes it look like he doesn't know what he's doing.
Mary Sue: A lot of people find Peter more interesting than Neal simply because Neal seems to have no significant faults. As an example, being a crack shot with a gun he's never handled before, despite being, quote, "not much of a gun guy."
Neal never specifically said that he hadn't handled the gun before, just that he didn't like it.
If you think Neal doesn't have faults after pretty much everything he does in the second half of "Point Blank"...
Forget "Point Blank", the entire second season shows how messed up Neal is. Then again, who wouldn't be after watching your girlfriend die in a fire.
Alongside plenty of the rest of season 2 after "Point Blank" has shown just how flawed and complex Neal really is, such as his wish to get away from crime and desires life more like Peter's. So, at this point, the trope has been subverted.
Alex seems like a Mary Sue as well: she's a gorgeous criminal with ties to Neal—the type of original character one would find in a fanfic.
OTOH, given that this is a cops-and-criminals show with the entirety of the female cast (and most of the male) being highly attractive, it seems to be par for the course.
Lampshaded slightly in "On Guard" when Peter keeps Neal's severed tie as a memento of, as Peter puts it, a rare moment of imperfection.
Never Live It Down: In "Forging Bonds", Burke becomes the butt of a number of jokes after inadvertently meeting Neal and letting him get away, and he keeps the lollipop Neal gave him as an eternal reminder of that fact. A more humorous one would be when Neal distributes pictures of Burke's mustache to his entire department.
Nightmare Fuel: The season 5 episode "Controlling Interest" has a psychologist who manipulates her patients into committing crimes. That's bad enough on it's own, but it gets really bad when Neal, while undercover as a patient, finds out he's been Drugged. You know things are bad when even Neal Caffrey is caught off guard.
One-Scene Wonder: In "Judgement Day", Bosch of Sterling-Bosch, who helps save Neal's bacon.
OT3: The fandom is fairly satisfied. Also averts Die for Our Ship, mostly due to Elizabeth's sheer awesomeness. The show itself also heaps piping-hot servings of OT 3 subtext onto our plates.
Some Anvils Need to Be Dropped: Most of the later Season 3 episodes have not been subtle in stating that despite the whole ankle bracelet thing, Neal's currently living the good life. He's no longer running from the FBI, has a job, a permanent home, and lots of new friends. And yet he's still willing to throw it all away for what he thinks he wants.
In his own words, Neal describes the con as an addiction or a rush. Only hitting rock bottom has a chance of changing someone from that life... and he (again in his own words) hasn't even hit rock bottom.
Special Effects Failure: The end of the first season finale, when Kate's plane explodes. The season two premiere creates one in the form of a horrible green screen when Peter and Elizabeth are eating. The green screen is also evident whenever they are talking over the phone. This was an attempt to hide Tiffani Thiessen's pregnancy.
Stalking Is Love: Peter uses his position to look at his wife's browsing history and states the Patriot Act gives him the right to do so. Neal correctly labels this as stalking. Later, it's revealed Peter had Elizabeth followed before he asked her out to see if she was dating anyone. Apparently, she quickly figured it out and held up a sign suggesting they go for Italian. The show trying to present them as an adorable, well-functioning couple and Peter as a largely by-the-book officer makes the the implications unfortunate.
Stoic Woobie: From the slight indications you get, her missing sister seems to have really screwed her up, but Sara is one of the most composed characters and sounds matter-of-fact even when she's being vulnerable.
Tear Jerker: Everything involving Peter and Elizabeth's marriage in "Veiled Threat". Sure, they make up, but watching a normally Happily Married pair have their relationship disintegrate through no fault of their own is heartbreaking.
Anytime Neal looks like he's about to break down and cry, which has happened twice now.
They Just Didn't Care: They mention Cape Verde was settled by the Portuguese, but still make it basically a Central American Island expy, complete with Spanish speaking populace, and a very conspicuous absence of people of African heritage (the actual island is off the African coast and the demographics reflect this).
Planning to profit from Nazi loot is okay if it's a Loveable Rogue doing it, apparently. Rather than, you know, trying to find the legitimate owners or their descendants. Eastin tried to ameliorate it by mentioning in his Twitter that the art came from museums. Because it's not like a good chunk of the art in the Reich's museums was appropriated/outright stolen from undesirables after the Nazi party came to power in 1933. Oh wait...
Mozzie mentioned that the art was from Russian museums.
Well, the show has been portraying the sale of the Nazi treasure as an ultimately morally bad thing, though not for the exact reasons stated above.