We are US citizens, cruising from Singapore to Shanghai. Want a few xtra days in Shanghai. Visa req?

Susan, if you enter and exit by the same cruise ship, no matter how many days you stay in Shanghai, you may need to apply for an entry permit in advance.
But if you enter by cruise and stay in Shanghai for no more than 144 hours, and then leave by flight to a third country or region rather than Singapore, you will be able to make use of the free transit in Shanghai. No entry permit is needed then.