What would happen if a client made out a cheque to an unregistered business name?

I don't know which country you are in so can't give a definites answer as I don't know how the law will affect this where you are. In the UK many small businesses are run with names that have not been registered with the authorities. As long as the bank knows that this is your "trade name" you should have no problem banking the cheque - indeed you may care to change the name on the bank account to J Bloggs trading as Joe Soap and then there would be no problem at all. Otherwise you may have to take evidence to show that the other name is your trade name. You should also of course let the tax authorities know your trading name.