Basic Trope: A detective — often a member of an official police force — who is lacking in skills of observation, usually to make them appropriate foils for an unofficial investigator.
Straight: Police officer Bob lacks the observational acuity and deductive reasoning skills possessed by Alice, an Amateur Sleuth, to the degree that Alice often proves helpful in assisting in his investigations, and Bob simultaneously is able to arrest the criminals whom Alice uncovers.
Exaggerated: Police officer Bob is so incompetent that he can barely function as a police officer, and is utterly reliant on Alice's skills in order to solve even the simplest of cases.
Downplayed: Police officer Bob relies on Alice's skills in this particular case but it is implied that he usually is quite competent.
Bob reached his position in the police force owing to internal politics, rather than deductive skill.
Bob is a reasonably intelligent and observant police officer, but Alice is a genius savant with observational powers that dwarf most others, including Bob.
Inverted: Bob is a competent police officer who is persistently tailed by Alice, an incompetent Amateur Sleuth whom he must help out of numerous scrapes.
Subverted: Bob appears to be an incompetent, but he actually uses Obfuscating Stupidity and is a lot savvier than he appears.
Doubly Subverted: Bob likes people to think that he uses Obfuscating Stupidity; actually, he's just that incompetent.
Parodied: Bob is utterly incapable of seeing even the simplest and most obvious of clues, even when someone is explicitly pointing them out to him.
Bob likes people to think he uses Obfuscating Stupidity, but he really is that stupid... except for a few specialized areas in which he is a genius... but Alice is still even more knowledgeable than him in those, so it's moot.
Alternately, Bob is perfectly capable of solving most standard crimes on his own but turns to Alice for help when he's confronted with the really weird cases.
Averted: Bob is a reasonably competent, savvy and observant police officer.
Enforced: "We want our detective to be free from the constraints that operating as part of a police force places on their agents, but at the same time being someone that they'd reasonably and credibly allow to be involved in the case despite being unofficial. We'd better make the police detectives less effective to compensate."
Lampshaded: "Sometimes, Inspector, it's a real mystery to me how you ever solve a crime by yourself."
Invoked: Amateur Sleuth Alice intentionally locates the dullest-witted police officer she can find in order to make her own skills more impressive in comparison.
Defied: "I'm not one of these slow-witted police officers you see in mystery novels, and I think you'll find not many cops are, either."
Discussed: "If we had cops like them around, crime in this city would be completely out of control."
Conversed: "How come the cops in these stories are always complete dimwits?"
Deconstructed: Bob's incompetence is indicative of a wide-spread paralysis, disarray and rot that exists within the police force, and is a symptom of wider issues of corruption, incompetence and even maliciousness which have resulted in several miscarriages of justice and criminals being allowed to escape justice, which in turn contribute to increasing social decay. In this situation, the public must turn to unofficial investigators in order to secure justice.
The police force has cleaned house to address the worst of such problems, but Bob is either sufficiently competent at his job to retain it or has otherwise hung on; he is among the last of a dying and unmourned breed.
Bob is a reasonably competent and intelligent detective; however, the crimes he faces are beyond even his abilities, while Alice is a super-genius.
Played For Laughs: Bob is the good-natured comic relief, who's bungling is a source of constant amusement to both the characters and the audience.
Played For Drama: Bob is exactly aware of how unintelligent he is and how Alice dwarfs his intelligence and skills, and resents it with a passion. As such, he obstructs Alice as much as possible out of spite, seriously damaging the progress of the case in the process.