Take the battery and get it load checked.   Don’t go chasing complicated stuff until you are SURE about the simple stuff.  Batteries can show good voltage but still not be able to start the car.
Check the firewall plug where the ignition wires go through the firewall near the accelerator linkage and near the driver side fender.

First, check the obvious, did you do anything recently before this issue started?  Did you unplug anything (like the neutral safety/reverse light plugs which are easy to swap and make it so the car won't start)? 

If that isn't the case, then I agree with MS, start with the battery.  Either get it load checked, or you can make sure its fully charged, then put a multimeter on it and have someone else turn the key and see what happens.  I've seen batteries that had 12.6V and dropped to 8V the instant the key was turned.

Next I'd check all cabling, particularly grounds.  I've seen bad cable ends cause a condition almost exactly like you describe.  If wiggled the car would magically start. 

After that I'd look at the solenoid.  Pretty easy to bench test with a multimeter and a good battery.  Should have continuity through it when power is applied to the S terminal, and no continuity without power applied.  If the solenoid is good then something is off in the ignition circuit wiring as BobE states.