Basic Trope: The police or other authorities figures are incompetent, skeptical of the main character's claims, and generally ill-equipped to handle the problems the characters are facing despite it nominally being their job.
Straight: Rachel has witnessed a murder and goes to the police to report it. The police are skeptical of her claim and, after a perfunctory investigation, refuse to do anything further, meaning that Rachel must uncover the murderer by herself.
Rachel is being hunted by a vicious murderer and frantically goes to the police for help. Despite copious amounts of evidence in her favour, the police officer assigned to her matter loudly ridicules the impossibility of her claims, calls over all his friends so that they can laugh at her, and then attempts to arrest her for wasting police time.
Same as above, but the police don't even bother to look at her, let alone talk to her, since they're all "too busy" eating donuts.
Subverted: Rachel goes to a police officer for help, not expecting him to believe her. The police officer takes her seriously and helps her.
Double Subverted: Turns out the police officer was just putting on an act for Rachel's benefit. As soon as she's gone, he throws her statement in the trash and ignores it.
Parodied: The police refuse to acknowledge and act upon a crime that's being committed right in front of them.
Zig Zagged: Rachel goes to report a crime to the police, only to be dismissed. After she's gone, however, the police officer finds that Rachel's report keeps nagging at him, and he goes through the records and discovers reports of a similar crime. Deciding that Rachel was telling the truth, the police officer acts upon Rachel's report in a competent and effective fashion. However, the threat that Rachel reported turns out to be more than the police officer can handle...
Rachel goes to report a crime to the police. The police immediately act on her report.
Suspecting that the police wouldn't believe her from the start, Rachel doesn't even bother reporting the matter.
The writer has a low opinion of the police force and this is reflected in the story.
The writer wants to write Rachel as an Amateur Sleuth, but needs to deal with the issue that the matter Rachel is investigating would normally be handled by the police.
Lampshaded: "The police in this city couldn't catch a cold."
Invoked: The murderer is a police officer who takes on Rachel's case so that he can bury it... and Rachel.
Exploited: The city's criminals dress up as paranormal creatures to commit their crimes, because any victims who attempt to get help will be dismissed as crazy by the police.
Rachel goes to report a vampire attack to the police, suspecting that they won't believe her. As soon as he hears her report, however, the police officer transfers her to the police department's Anti-Paranormal Threat Taskforce, which they have set up for just this eventuality.
The city has passed an ordinance meaning that the police must investigate every matter that is put before them, no matter what.
Maybe the desk cop doesn't believe the "vampire" part of Rachel's claim, but she's obviously distressed, something bad happened, and the "attack" and "tried to kill" part of her story are worth investigating.
Rachel was smart enough to instead report, "There's some nutjob dressed up like Dracula going around attacking people." The police would consider "a dangerous and delusional person" more plausible that "a real-life vampire".
Discussed: "You don't report vampires to the police; everyone knows they won't believe you."
Conversed: "It's a wonder that the city in this story hasn't completely burnt down already, given how incompetent the cops are."
Deconstructed: The failure of the police to act on Rachel's report means that a criminal gets away with a crime. Rachel is murdered by the criminal to keep her silent.
Reconstructed: It later turns out that Rachel faked her death in order to call attention to the incompetence of the law enforcement system. This allows a new commissioner to be elected to clean things up and get personnel to take crime reports more seriously.