On a separate point maybe even be able to put people on trial who are in the dungeons to determine if they are a traitor or not.
That could turn out to be really interesting. Some of the most severe crimes could warrant a direct execution, while some murkier ones would need a court case to decide the guilt. If done properly, the character aspect could be very deep (which is what the devs are going for).
Also, if a character inherits a title as a result of a suspicious death, then they might have to solve the murder, or suffer an opinion penalty with vassals if they don't. After all, would you trust your liege, if he/she can't even find who killed the previous ruler? This could be limited to King/Emperor and it could possibly only effect the vassals of the gained title. This means that you'll be pressured to find a culprit. If you can't find the actual culprit (or you are the culprit), you could try to accuse somebody else. This should only work if you have very high intrigue and/or diplomacy, in comparison to the accused. Accusing an innocent landless character who doesn't have anybody to defend them should be much easier than a powerful and influential vassal (or even an enemy King). A high religious character might be able to convince that a suspicious death was an "act of God", instead of having to find a culprit. Finally, a high martial character might for for trial by combat, if laws allow for it.
Oh yes, I suppose I should add that clarification: plotting to assassinate other nobles in the same realm should be a justification for imprisonment, etc. Targeting other realms could fall either way. I'd argue definitely little qualm if it was a realm of another religion. Same religion makes it murkier.
A court case could be the way decide this. If it's a clear crime, then execution is almost a certainty. However, if the defendant can mount a proper defence (e.g. victim was heathen, has higher diplomacy/intrigue than accuser, the realm prefers him/her to the accuser, ect..) then it could be a risk failure could be bigger. If the accuser looses the case, then they should loose face (opinion loss, or if the case goes extremely badly, they could be accused themselves, especially if they are the real culprit).
If you want to avoid a case, then you could jump directly to the punishment, as it might be too much micro for every crime. The punishments could be fines, loss of limb, being sent to a monastery, imprisonment, execution, ect... As a guidance, a list of crimes and the appropriate punishments would be required. Tyranny would be incurred, if the culprit isn't certain and/or the punishment is too sever/soft (only cutting the culprits had off for regicide could anger vassals as much as being too harsh). The punishment could also be linked to the judges type of character and possibly using the stress mechanic.
If one vassal accuses another within the same realm, then their liege should reside/be judge, while if you are accusing a subject you would be the judge (unless a law is introduced, creating a high judge...). If the defendant and accuser are in different realms, then an independent adjudicator will be required. If they are they are the same faith, this could be a neighbouring religious authority. The Pope might even be called upon, if Kings/Emperors start accusing each other. I'm not really sure who it should be, if the accuser and defender are in different realms and of different religious (a simple solution could be to just spawn a random character).
TLDR; Crimes should have matching punishments, depending on the severity. If you want to accuse somebody who's guilt isn't certain; impose a disproportionate punishment without tyranny; or accuse somebody who isn't your subject, then you will need a court case. The outcome should depend on the diplomacy/intrigue of the accuser and defender, relationships, as well as the actual evidence (possibly coming from evidence/'hook' system already announced). A case should be a risk, as if you loose, you will have to suffer the consequences, ranging from opinion loss to death.
