'No eating or drinking' , 'no eating and drinking', which is the correct way of saying?

I would assume that the word "allowed" would be understood as with a 'No Smoking' sign.

To say "No eating and drinking" would mean that someone who is doing both activities at the same time is NOT allowed. Therefore, if a person was only eating, that would be acceptable. The same for someone who was only drinking.

On the other hand, "No Eating or Drinking" would prohibit both activities either separately or simultaneously.

Neither is really correct:
- Eating and Drinking is forbidden
would get the message across with no miscommunication.
OR
- Eating or Drinking is not allowed at this location.

sometimes you have to hit them on the head........

The correct wording is "No eating or drinking". "OR" is the correct connective word in this context.

This would be for a sign, such as in a library or a dress shop, therefore it does not need to be a complete sentence.

I prefer "or" because and could be interpreted that only the two, when both done at same time, is forbidden. We all know that whether "and" or "or" is used, both are forbidden.

However, sometimes it does matter: No drinking and driving means, specifically, that doing both at the same time is forbidden, but neither, when alone, is forbidden. We do not say "no drinking or driving" after all.

Strictly speaking, "No anything" is not truly a sentence. No eating or drinking is allowed. neither eating nor drinking are allowed. (Both) Eating and drinking are forbidden.

No eating and drinking