Shower head not working
What is (very probably) wrong is that the shower is on an outside wall and poorly insulated, so it's frozen. Run straight hot water through the hand-held that is partly working before it can freeze solid, and that will bring some warmth to the affected pipe, hopefully helping it to thaw. Heating the room more (turn up thermostat) can also help. If you choose to use a space heater to help warm the room, be very, very careful (frozen waterline is much less trouble than burning the house down.)

You might also try using a hair dryer on the shower control and the pipe for the showerhead (idea being to warm those parts and help thaw parts frozen inside the wall) - don't let it get wet, of course.

Once you get it thawed, you should probably leave it "dripping" as if there was a slight leak until the weather warms to temperatures where you have not had this problem before. The flow of water will help prevent the formation of a complete ice blockage.