In the past, conviction of a crime for a noble or aristocrat would result in, at best, heavy fines, reparations, and dishonor on the family name or house. At worst, loss of wealth, lands, and title could result. Execution would be for the absolute worst offenses.
In Noble society, reputation is everything. You won't want alliances with a house that's members have committed crimes. I think the offences should be discoverable by other players interacting with members of that house for a period of time. This will add a social element that will necessitate the head of the house to punish criminal behavior beyond what the law would. I think it would enrich the social environment.
For everyday folks, loss of property, fines, jail, corporal punishment, public humiliation were the norm. How about the stocks? The character is held in an area of the town square for a length of time to be humiliated by those who pass by. Their crime posted above their head. Maybe a rotten fruit throwing mini-game? The offender is forced into a public "greased pig catching" mini-game to get free early or reduce debuffs? Lots of fun things you could do with that. Again, it enriches the social environment and makes the game world more believable while discouraging crime.
Not sure how execution would fit into the game system, maybe for confirmed cheaters/botters? This could result in a perma-death and/or account ban? Preferably an in-game public execution to discourage other offenders. Again, something that was done to criminals in the past.
I'm sure someone has mentioned this before. But features like this would be both interesting and beneficial to the game, while staying true to the setting.