What could be wrong if the speedometer stopped working and the transmission will not shift?

It sounds like the VSS (Vehicle Speed Sensor) has failed. If the VSS fails, it can cause the symptoms you are describing. Check the wiring and make sure your VSS is properly plugged in before you replace the VSS. Where the VSS is located can vary depending on Make Model and Year. Check your vehicles Service Manual (Available from auto parts stores, commonly referred to as a "Chiltons Manual" or "Haynes Manual". ALLDATADIY is another good source) for more information.

Some vehicles do not have a Vehicle Speed Sensor, a common example would be a Generation 3 Ford Taurus SHO. The SHO uses the ABS wheel sensors to generate a signal to the computer to calculate speed. If this is the case with your vehicle, it is recommended you take your vehicle to reputable shop for servicing, as the ABS system is a complex and important safety system for your vehicles safe operation.