Technically yes. For DSS and private school offers your son received, you can register them all. (Note that individual school may even warn the parents that they will cancel their child's seat once they knew in whatever means that the candidate has took up other school's offer.)
Yet, please be considerate as we all understand how we feel while waiting for an offer. Once decided to take a new offer, please do release the previous offer timely by informing the related school explicitly instead of no show until next Jun or even Sep.