There's a few things to consider. The neutral safety switch can be intermittent. Try shifting between "park" and "neutral" a few times, and try starting it in neutral.
Some starter solenoids will not engage far enough to turn the starter motor on if battery voltage is just a little low. When the starter doesn't engage, try to not disturb anything else that might make it crank, then connect a battery charger, then try starting it. If that works, suspect the battery or the cable connections.
We had a '98 GMC Lemon Law buyback donated to my college by GM that I used a lot to teach electrical diagnosis. One of the biggest problems we ran into that wasn't built in by me was stretched terminals in the starter relay socket in the fuse box under the hood. In spite of being warned, students would poke the voltmeter probe into those terminals and stretch them. Meter probes are much fatter than the relay's terminals. The symptom was there might or might not be a very light clicking of that relay when turning the ignition switch to crank", but the starter solenoid wouldn't engage. Having a helper press in the relay often would get the engine started.
If nothing else works, remove the starter relay, then use a stretched out paper clip or piece of wire to jump two terminals. Whether or not the starter cranks will tell us which part of the system is working. Do this with the ignition switch off. Jump two terminals together in diagonally-opposite corners of the socket. If nothing happens, use the other two terminals.
Wednesday, October 28th, 2015 AT 1:49 PM