I have an older Murray 40" riding mower and the ignition switch stopped working. Sometimes I can jiggle the clutch and it will turn over but now it does nothing.

Hi,
Look under the seat for the model number, it begins with the deck size, i.e. 40. I will give you a link to a parts breakdown, most of which include the electrical schematic. From your description I would say that the switch associated with the clutch, the one you jiggled, is likely bad since the battery voltage normally goes through two safety switches to reach the trigger circuit on the solenoid.
Peace,
Clarke