@Puffer Fish
For example, if the only evidence is the woman's claim, should that be enough to be able to convict? The jury does not know because there is no law to guide them.
Normally her claim would not be enough to convict and this is clear in all criminal cases. It is a bit complicated but here is what US law says. It is not on the wife to prove anything though she may be the initiator of the allegations against the defendant; in this case her husband. She makes a claim and the state investigates the claim. If they believe that a crime was committed and that they have enough evidence to go to trial they go to a judge to get a warrant for the husbands arrest.
So. When the trial begins, it is the state who is attempting to prove the husband's guilt and the wife, though the alleged victim, is really only a witness to the alleged crime. The defendant does not have to present any evidence at all of his innocence and usually will not unless it shows clearly that he did not do it. For example, evidence that he was elsewhere and could not have committed the crime. He need not even testify and, in a case like this, very likely will not testify.
Before the trial begins and again before deliberations, the judge instructs the jury that in order to find the defendant guilty, the state must prove that he (in this case raped his wife)
beyond a reasonable doubt. This means that the prosecution must convince the jury that there is
no other reasonable explanation that can come from the evidence presented at trial. In other words, the jury must be virtually certain of the defendant’s guilt in order to render a guilty verdict. The state's case would fall apart and a not guilty verdict result if this standard is not met. In the case of "he said, she said" the jury would be forced to acquit. In fact, going back to the initial decision by the state to pursue charges, a case of "he said, she said" should never have resulted in a warrant for arrest because the allegations would not have been supported by another standard called "reasonable grounds to believe that a crime had been committed". But lets ignore that for the moment.
If, during trial, the evidence presented by the prosecution (state) does not meet the standard of beyond a reasonable doubt, and that is clear to the judge, the judge can either direct a verdict of not guilty or dismiss the charges without a verdict which stands, legally, as not guilty. If the judge believes the case should go to the jury, he gives detailed instructions to the jury about the standard for conviction, how to consider evidence and other considerations under the law. Then the jury must all agree that the defendant is guilty or not guilty. Anything less than this results in a "hung jury" and will, at the judge's discretion either cause him to dismiss the charges or order a retrial with a new jury. As a practical matter, in jury deliberations when jurors disagree the verdict will eventually be not guilty. But remember. The defendant does NOT have to prove his innocence. He is presumed innocent until proven guilty by the state.
I will mention at this point that in a civil case, the burden of proof is "preponderance of the evidence" not "beyond a reasonable doubt" so if the wife sued him for this rather than participating in a criminal trial the road to getting a decision in her favor would have been easier but fraught with danger lest he win and she get stuck with the bill. But then I digress. You are talking about a criminal case.
Or in another case, if some illegal object is found in a person's house, should that be enough to convict them of an illegal possession charge? Again, the law does not give any guidance on this to a jury. The jury is pretty much going to have to make up their own law, because the law does not tell them how to interpret it, in what are very common types of situations.
The law does give guidance for this.
In the first place, the way the object is discovered is key to even bringing charges. If the police wish to search for something in someone's house they almost invariably must get a warrant. The state must go to a judge, explain what they are looking for and why they are looking for it. They must further present to that judge evidence that they have to make them believe that this "object" is in the house and that it belongs to the person who owns the house or is there with the knowledge of the person who owns the house. If they can't do this, the judge will decline to give them a search warrant and they may not even enter the house. The police can't get a warrant and go on a fishing expedition to see if anything illegal is in the house. They must be clear about what they are searching for.
It is possible that while conduction a legal search, lets say, for example, for a stolen gun, the police find illegal drugs. This may or may not result in a drug charge. That is up to the judge to determine. If the police enter someone's house without a warrant and find, for example, a stolen gun, then the judge will rule that the evidence was illegally obtained and rule out the evidence so no charges would result. (There are exceptions but this is the principle.)
At least under US law, the protections you are concerned about are in place long before any one event triggers the state to act against a citizen so they need not be explained in every law. Some laws, however, do contain elements of evidence as part of the law. It is just does not need to be very detailed because the rules of evidence are already clearly established under US law.
Does this explain the situation under US law in this case?