Is everything one says to a cop in the course of an enquiry really admissable as evidence against them? Even if they deny allegations? Why is that?
It is somewhat understated, because your silence can also be used against you. In Salinas v. Texas, defendant Salinas was "just talking" to police, not in custody, and his silence (as opposed to shock and outrage) at the question of whether shell casings found at the crime scene would match his shotgun. This "adoptive admission" was introduced as evidence against him. In order to prevent your silence from being used against you, you must invoke the 5th amendment. There are two exceptions to the rule that you must invoke the 5th to be protected by it. First, a criminal defendant need not take the stand and assert the privilege at their own trial. Second, a witness’s failure to invoke the privilege must be excused where governmental coercion makes their forfeiture of the privilege involuntary (Miranda). In lieu of coercion, you must preface your silence with an invocation of your 5th amendment rights.
The Miranda Warning doesn't go far enough. It clearly states that you have the right to remain silent, and anything you do say can be used against you. The statement should include the fact that you must assert your Fifth Amendment rights using the exact correct phrasing, that saying anything after the Fifth Amendment rights are invoked can invalidate the Fifth Amendment assertion, and that saying anything that may be helpful towards your case may be dismissed in court as "hearsay." Law enforcement officers cannot be helpful to your case in any way, so saying things that you may think are helpful to your case will not save you in court. For more information, I recommend watching an informative video I found called Don't Talk to the Police.
This depends on the relations between the police and community in your area.
Every interaction between the police and the public is, first and foremost, a social interaction between individuals. Only a small percentage of interactions leads to charges being filed, even if the member of the public has recently done things that could lead to charges. A strategy to optimize the likelihood of a positive outcome in court might also increase the likelihood that the case will go to court to start with.
Officer: "Do you know your tail light is busted?"
Motorist: "I take the Fifth."
Officer: "Do you know your tail light is busted?"
Motorist: "Last week it was OK. Thanks for telling me, I'll have it fixed."
When there are good relations between the police and community, the second motorist has a much better chance to be let off with a warning. Or perhaps a fine in addition to the warning. The first motorist would make the officer wonder why a driver would invoke the Fifth Amendment over an obvious technical fault. That might make the officer look for reasons to justify a search of the vehicle.
Officer: "Did you shoot the victim?"
Gunman: "I take the Fifth."
Officer: "Did you shoot the victim?"
Gunman: "He was threatening me."
In a case like that, there is a high likelihood that the gunman will go to court, and the stakes are enormous. "Lawyering up" is probably a very good strategy, because any statement to the facts could be misinterpreted or taken out of context.
The statement is accurate in one sense: statements made out of court by a party to litigation (including by a criminal defendant) can generally be introduced at trial by the opposing party despite the hearsay nature of the statements. This is known as the "party admission" hearsay exception. See R. v. Foreman, 2002 CanLII 6305 (Ont. C.A.) at para. 37.
But there are still limits to the admissibility of that evidence.
First, the statement would have to be relevant, but I assume this is not the aspect you're interested in.
The bigger concern is voluntariness. Statements given in custody carry the heightened risk that they are a false confession: "A large body of literature has developed documenting hundreds of cases where confessions have been proven false by DNA evidence, subsequent confessions by the true perpetrator, and other such independent sources of evidence" (R. v. Oickle, 2000 SCC 38 at para. 35). False confessions "almost always involve 'shoddy police practice and/or police criminality" (para. 45)
Thus, in Canada:
a confession will not be admissible if it is made under circumstances that raise a reasonable doubt as to voluntariness. ... If the police interrogators subject the suspect to utterly intolerable conditions, or if they offer inducements strong enough to produce an unreliable confession, the trial judge should exclude it. Between these two extremes, oppressive conditions and inducements can operate together to exclude confessions (para. 68).
It's completely wrong
The cops can do nothing but detain you, they are the executive. But they will record whatever you say that is not to your lawyer for someone else: the state attorney.
And the state attorney can use everything you say against you in the trial if they believe it will further their case against you.
The reason for this is because of the evidentiary rules of U.S. Court. Out of court statements made by someone other than the person testifying as a witness are not permitted under the rules of hearsay with a few exceptions. One of those exceptions is a "Statement against adversarial interests." That is, if your adversary witnesses Alice said something to Bob that would harm Alice's interests, Bob may testify to what Alice said because Alice is not likely to testify to it in court.
Let's say that Alice invokes the "Marley Defense" when accused of shooting two law enforcement officers ("I shot the sheriff, but I did not shoot the deputy.") to Officer Bob. This in fact, contains two statements by Alice of (1.) Alice shot the Sheriff AND (2.) Alice did not shoot a deputy. Statement (1.) is an admission of guilt and would be against Alice's interests in court, thus Officer Bob can testify that Alice told him she shot the Sheriff because it can be assumed Alice would not openly say that in court because it is an admission of guilt. But statement (2.) is a defense against guilt and thus it is a Statement in favor of Alice's interests (here being sentenced for the crime she did commit, not the one she didn't commit) so Officer Bob testifies to this in court, it will be a violation of hearsay and would be objectionable (Ironically, the Prosecutor would object to this answer because it hurts the prosecution. The defense would be all to happy to let it slide. And yes, witness testimony itself can be objected too.).
Even if Officer Bob believed both of Alice's statements to be true, he can not make hearsay statements. It's on the defense to prove that Alice did not shoot the deputy.
I am aware that the tag on the question indicates the asker is from the US, but for the wider audience this may be of benefit. In the UK, in many circumstances if you do not tell an officer your name, address and date of birth, they may arrest you for the simple practical reason that you might be difficult to locate again later. So it is sometimes in your interest to give that information when asked. Some people believe it's better to wait until after you have been booked, I don't have the expertises to make a strong statement on that, but personally these are questions I would always answer when asked to reduce the chance of being arrested in the first place.
Furthermore, when interacting with the police;
You do not have to say anything, but, it may harm your defence if you do not mention when questioned something which you later rely on in court.
This is further discussed in this Wikipedia article;
The Criminal Justice and Public Order Act 1994 provides statutory rules under which adverse inferences may be drawn from silence.
Adverse inferences may be drawn in certain circumstances where before or on being charged, the accused:
fails to mention any fact which he later relies upon and which in the circumstances at the time the accused could reasonably be expected to mention;
fails to give evidence at trial or answer any question;
fails to account on arrest for objects, substances or marks on his person, clothing or footwear, in his possession, or in the place where he is arrested; or
fails to account on arrest for his presence at a place.
So, in short, if you are accused of a crime, and have a convincing defence such as an alibi, it may cause problems if you refuse to share it before going to court. I believe that this is mostly to avoid wasting court time. You are entitled to legal representation when being questioned though, so the legal representation would presumably advise you on this point.