Our hampton bay ceiling fan works fine but the light under the fan does not. I changed the bulb but it still does not work. We use the wall switches, not a remote.

Did you test for power on the incoming wires for the light fixture, usually the blue and the white.

At the double switch on the wall there is a single black and a single white wire coming to the switch from a rats nest of blacks and whites deeper inside the box. There is power at the switch according to my tester and of course the second switch controls the fan, which is working.
I took off the ballast inside the fixture and tested the lack and white going from the fixture to the ballast there, but there is no power either with the wall switch on or off.

Are you saying your have a switch with a white wire and a black wire go to or coming from it?

Are you saying your have a switch with a white wire and a black wire go to or coming from it?
Yes, I have a fan/light on the ceiling, and on the wall there is a dual control (double pole combo switch) that has one black and one white wire coming to the switch as pigtails from wire nuts with several wires in them further back in the switchbox.

Your switch is wired wrong. Black is hot and white is neutral, flipping your switch should cause a dead short. A switch only interrupts a continueous black.

You may have a switch loop in the box, if a white is on the switch, it should have been reidentified with black tape.
With no tape, it is easy and dangerous to get confused with a neutral.
If not to tough, maybe remove the fan and connect 2 sockets with bulbs to see when each is hot?