The doorbell button is usually the culprit when bell or buzzer trouble occurs. It is always wise to check the button first if a doorbell or chime fails to function. Metal fatigue in the spring or corrosion from the weather may have caused the contact point to fail to function.
If the trouble is not in the bell button, check all visible sections of the wiring for breakage or mechanical damage.
Use a low voltage tester to test each section of the wiring. You are more likely to locate a break in the wiring than a short.
If the bell continues to ring even though the button is not pushed, you probably have a short somewhere in the wire.
If the transformer has an overload protection device, a shortage many cause it to cut off the secondary current. This means the transformer should also be checked.
Disconnect one wire from the transfomer's secondary screw. Touch the low voltage terminal and the disconnected wire with the low voltage tester.
If there is a short, the bulb will light. Be sure the bulb you are using in the low voltage tester mantches the voltage in the transformer. If you are using a lower voltage bulb it will burn out. If you are using a bulb with a high voltage rating, the light will simply be dim.
You can use a low voltage connector to test the doorbell button also. Connect it from one transformer terminal through the bulb tester and back to the other terminal. If the button is working, the bulb will light up then the button is pushed.
If the trouble seems to be in the bell, buzzer, or chime, it is usually the connections at the contact point. Remove each connection wire, file it with sandpaper and replace.