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.
Harsher in Hindsight: Ads in this film seem to know the targets' names, personal tastes, and the like. Come the age of selling users' personal information to third-party companies for advertising purposes...
Idiot Plot: There really is no good reason for Anderton to go after Leo Crow. Avoiding him until the time of his murder was over and returning to find out what happened later would have been a more sensible thing to do. Also, Lamar likely could have easily killed Anne Lively outside of Pre Crime's jurisdiction, making his need to try and fool the system unnecessary.
Yeah, not going after Crow might have been more sensible - but we are speaking with the benefit of hindsight here. Caught up in the heat of the moment, Anderton's judgement is clouded by doubt, panic, curiosity and disbelief. As Witwer points out, Anderton won't just run and hide, "because he thinks he's innocent". Even when Agatha tries to get Anderton to turn away from his "destiny", Anderton won't, as he is so driven to find out what is happening.
Magnificent Bastard: Lamar Burgess. He literally made a career out of faking out Precrime; first by disguising Anne Lively's murder as an "echo", then by disguising all three visions of Crow's death as brown balls by putting the plan in action while Anderton was at the office, ensuring that he would either be arrested immediately or run, so when Crow be found with the Orgy of Evidence that could lead Anderton to murder it looked planned by Anderton. The only thing Burgess couldn't see coming was Anderton figuring out everything in time to tell it all to his wife, even if he wasn't free to act on the information himself.
Strawman Has a Point: Really the only problem with precrime is the fact that Director Burgess murdered Anne Lively instead of taking it the courts and leaving it in their hands if the woman should be given her daughter back. Outside of a complicated coverup to murder Lively and the questionable treatment of the precogs, precrime really does work.
Precrime might work, but the sentencing procedure is still all wrong. People are being jailed for murder without the possibility of parole, effectively having been given a Fate Worse than Death, with the justification that because they did commit the crime in future, they are guilty of it. Despite the fact that by preventing them from ever having committed the crime, they haven't actually done anything illegal, yet still treated as if they had. In fact, they claim that it's because You Can't Fight Fate, yet if that was true, the murders would still happen?! They fight fate all the time! All in all, the most you could convict someone of is either attempted murder or conspiracy to commit murder, and that's disregarding the probably-numerous cases of people who had merely given little or even no thought to killing victims they may not have even met yet.
It's a science-fictional extrapolation of current criminal investigation techniques; a few decades ago, who would have predicted that people would be convicted based on microscopic organic residue, or evidence reconstructed from fragments of years-buried human bone? The precogs are simply another unexpected source of evidence. In the "brown ball" prevision arrest, Anderton shows up less than a second before the murder is committed. The commercials for Precrime show that this is not an isolated incident - the precops are used to yanking weapons from murderers' hands right in front of their victims. Changing the future does not change the intent behind people's actions. The reason Precrime is abandoned is not because it doesn't work, but because of the cost to innocent lives; the Minority Reports and the precogs themselves.
The heart of the film runs on this as illustrated by the arguments by John (for it) and Danny (against it) in the first act, and that the film doesn't present either person as completely right or wrong.
The Woobie: The three Precogs. They were born to Fantastic Drug addicts and spent all of their lives having nightmares about murders. Then, people started to get wind of this and whisked them off and did unmentioned (but implied to be nasty) things to them. Then the Precog system was set up, which meant that they were forced to spend years and years lying in a drugged stupor in a pool, watching endless future murders. Oh, and the one time we hear about one of the parents trying to save her child from that, she's murdered. And said child is forced to watch her own mother's death and then watch as everyone ignores it.