1996 Cadillac Deville

If the connections are clean, tight, and corrosion free, then check for signal to the starter by disconnecting the two wire connector at the front of the rear bank valve cover and checking for 12 volts, or a test light to light when in crank position. The signal wire is large purple wire, and make sure you are at harness side, not the starter side. If it has a double connector end on the positive post, there should be a lead spacer in between. If it is not there the starter will in most cases, not engage.

It could be a fuse, it could be a park/neutral switch, ignition switch, starter, several things. If you can verify whether or not a signal is getting to the starter, we can eliminate all except the starter. You said it clicks, does it click once, each time the crank position is engaged? Does it make several rapid clicks at that time, almost a buzz. What do the lights and accessories do at that time?

Does the click come from under the hood, once every time crank is reached? It sounds like a starter to me, but that is based on our discussion.