Get out your multi meter and measure the voltage output at the terminal strip of the controller. One lead on the common and the other lead on station 1, then 2 and so forth. Each station, when it is turned on, should have 24 volts AC.
Check the resistance through the valve solenoids. Do not have any stations on when you do this. Put the multi meter on ohms and put one lead on the common and the other on station 1, then 2 and so forth. The resistance through most solenoids should read between 25-50 ohms. Lower readings indicate a short circuit and higher readings indicate an open circuit.
Go to the valves with your multi meter. Disconnect the wires at the solenoid. Put one lead on the common, usually the white wire and the other lead on the hot wire. Turn the station on at the controller and measure the voltage. If it is 0 volts then you have a broken wire. If it is 24 volts then connect it back to the solenoid and check the voltage. If it dropped to 0 volts you have a ground short. Either way, you find the wire break/ground short and repair it or replace the wire from the controller to the valve.
When the heads on the one station pop up briefly and then drop down is there water still running through the valve? With the station on, touch the valve and listen. Do you fell or hear water running? If yes look for a broken pipe and repair it.
Most 'home owner' valves are inexpensive and the valve is cheaper than the parts. If the valve needs to be rebuilt, buy the exact same valve, gut it for the parts and rebuild it.