My car suddenly will no longer start. The starter is not engaging the flywheel.
Battery is new, all the instrument lights come on when I turn the ignition switch
to position 2, but nothing happens after that except a click. I have to pop start
it to get it going. I checked all the fuses, and they all look fine. What would
cause this to suddenly happen? Never had it happen before.

 

So by 'pop start it' you mean roll it down a hill and spin the motor using 2nd gear on your clutch car?

Sounds like a replacement starter is needed. Turning the ignition and getting only a click is a likely indicator; this symptom will happen a couple more times in your life, just remember it - no sweat. You can also get someone's help to hit the starter remotely (using a big hammer), force transmitted via long stout metal rod or broomstick, while he holds the key in the start position. It'll start immediately with the hit and that means new starter needed = definitive test.

I would get starter from NAPA, dealer or maybe RockAuto, avoiding Oreily's or AutoZone or like.

 

I'd also recommend a Denso starter, because the aftermarket brands can be hit an miss.

The alternative is to replace the solenoid contacts and solenoid plunger. Over time the plunger digs a step into the contacts, where no electrical connection is made when you hear that click. I did this on my 95 manual transmission starter and the repair lasted over 8 years. I think I had to source them from ebay at the time.

 

Thread title says "no signal to starter" and that's another possibility.

Did you check the small wire to the starter?
Make sure that it gets battery voltage when you turn the key to Start. (Be careful not to short the BIG wire at the starter.) If that small wire doesn't get power when you try to start, then the problem is in the ignition switch or wiring, and not a problem with the starter.