Which No. is valid? Sailno. or No, graved into transom?

I am not sure, but since you can put any Mirror sail on your boat that you want, I would guess the transom number is the number of your boat. Certainly, if you want to know when the kit was constructed you would look up the transom number. On the other hand I think in a regatta your sail number would be noted.
Pete

It's definitely the hull number that counts. The sail number is used when racing because it's the only one that can be read at a distance, but the boat is the hull and not the sails. It's common for sails to be sold on by more serious racers when they replace them with new ones, so it isn't impossible for more than one boat to have the same sail number on them in a race, in which case the hull colour or crew's clothing would need to be used to tell them apart. That shows you why a register of boats based on sail numbers wouldn't be greatly useful. It is also possible for two hulls to have the same number as well, but that only happens when someone replaces a hull with a new one of the same number (while the old one is no longer officially a Mirror and can be blocked from entering races) - I don't know if that is still allowed, but it certainly did happen in the past: there were two 15345 hulls, for example, and I sailed the older one for several years after the plan to burn it was cancelled.

Welcome to the forum!
Your question has been answered, I see: the hull number identifies the boat!
Are you in NL or DE ?
Cheers and happy sailing.

Gernot H.