Dodge Intrepid won't start sometimes!

I have a 1997 Dodge Intrepid, and once in awhile it won't start. When you turn the key, it just clicks (one click only for each time key is turned). Usually will start if you jiggle the gearshift. Otherwise can start it in neutral. This is an intermittent problem, and there does NOT seem to be a pattern. I can go weeks with no problem, then have it two days in a row. I was thinking maybe some sort of neutral safety switch problem? But does my car have one? Where is it? And how would I test it?? Any suggestions/ideas appreciated!!

every automatic transmission has that safety switch, located next to the valve body.

Thanks, Jeremy06! I'll give it a try this weekend.