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.
Alternate Character Interpretation: There are two competing alternate theories about Columbo (other than Obfuscating Stupidity). One is that he's a Genius Ditz. The other is that he isn't smart at all, just an average cop. He himself claims in one episode that the main reason he's successful is that he's a professional with years of experience in hunting murderers, while most of the murderers he captures are amateurs who are doing it for the first time and thus making rookie mistakes.
Either that, or (as he freely admits) Mrs. Columbo solves all the crimes when they talk about it over dinner (which he probably cooks, given that he's shown quite a proficiency and interest in cooking on a number of occasions). Assuming she exists (Falk himself once claimed that Mrs Columbo is Columbo's way of being nice to the criminals with the added get out clause should someone claim conflict of interests that "it's not me, it's my wife who likes them").
Is Columbo a Manipulative Bastard? He's overly nice to people in a bloodhound sort of way; he convinces people that he's just a country bumpkin more interested in whatever 'hat' the villain wears than solving the crime, only to reveal in the end a cold detachment and clinical mind that the bumpkin persona allowed free reign. He plays with the feelings of the criminals, making them like him (more often than not) or at least pity him and drop their guard, or he pushes them subtly and continuously to the point where they break.
Peter Falk gives his take in his memoirs "Just One More Thing". In it he says Columbo is absent minded, but that's because he's concentrating all his thoughts on cases he hasn't solved.
Sometimes, he looks like a troll — plainly trying to annoy and distract the suspects so they may be provoked into blurting things he would never get via normal investigation.
There are some clear hints that, while he might not exactly be a literal genius and might be genuinely absent-minded and eccentric, at least some of it is for show or played up in order to throw the killer off guard. For example, in "Death Lends A Hand", when Columbo is only around the victim's husband — who he does not suspect is the killer — he acts in a professional and reasonably intelligent fashion. It's only when the actual killer (a private investigator hired by the husband) shows up, and Columbo gets a reason to suspect him (the ring the killer wears, which matches a cut on the victim's face from when he struck her) that he starts to act the clumsy oaf.
"Funny Aneurysm" Moment: Throughout the series, Columbo expresses a severe dislike of dentists, and the few times he has to go it ends up being a total nightmare for him. It's believed that a bad reaction to drugs during a dental visit is what led to the onset of Peter Falk's dementia, the illness that killed him.
Two actors who played murder victims later became victims of actual murder- Barbara Colby("Murder By The Book") and Sal Mineo("A Case of Immunity")
Hilarious in Hindsight: "Etude In Black" is the same episode in which Columbo gets Dog. Trying to think of a name and hearing that the victim named her bird Chopin, he considers naming his dog Beethoven.
Magnificent Bastard: Columbo himself, as well many of the killers (notably the ones portrayed by McGoohan and Cassidy).
Padding: Many of the two-hour episodes suffer noticeably from this; since the Lieutenant didn't have a personal life by conceptual mandate, the writers were forced to stuff in scenes like him taking the dog to the vet or asking a suspect where he'd bought his shoes.
The Problem with Licensed Games: In a rare example predating video games, there was a Columbo board game. The problem was, it really had little to do with the character and the only image of Columbo was a simple drawing showing him from the back, one can assume because they didn't have permission to use Peter Falk's likeness.
They Wasted a Perfectly Good Plot: "Rest in Peace, Mrs. Columbo" involved Vivian Dimitri, the ex-wife of Pete Garibaldi, a man Columbo had arrested years before, plotting to kill Columbo's wife because her husband died in prison of a heart attack, for which she blames Columbo, as well as her husband's partner Charlie Chambers. It's too bad that the case in question was never one that had been filmed as an episode.
The Woobie: Numerous murderers are pushed around by Asshole Victims, such as Beth from "Lady in Waiting" who is held down by her dominating brother and mother for her whole life. Once her brother is dead her life turns around and she's suddenly more assertive and in control of her life for once. She later takes it to the point of becoming a Jerkass Woobie, or just a plain Jerkass, however.