Silicone is nit the answer... cure the disease don't treat the symptom.

Since it's cheap and easy to do... replace the fill valve. You might have a hairline crack in the threads that only opens when tightened.

And don't overlook the supply line.

Correct, replace the fill-valve, and do not over tighten the supply line.

Thanks for responding, I'll try replacing the fill valve.

I have never touched this toilet in the last 15 years to tighten the locknut or anything else, so I'm puzzled how this leak could have started. As I said before, I cannot see any water drops except under the plastic locknut, regardless of how tight the locknut is.

You have to start at square one and eliminate things one at a time. Start with the cheapest and easiest... the fill valve cause it will come with a new gasket. Then the supply line.

OK, so I installed a FluidMaster 400A fill valve, and the leak stopped! But now, every 10-20 minutes, it will flush for 5 seconds - I guess this is called "ghost flushing". I didn't have this problem before.

I cannot see water leaking into the toilet bowl, but have read online more than a few people saying this problem started once they installed the FluidMaster. Besides replacing the flapper, is there anything else I can do to fix this?

Lift the flapper and verify that the flush valve is smooth and that the flapper does not have a piece of debris dislodged from the water line when the other repairs were made.

You may end up changing the flush valve, as well.

More than likely it is the flapper.

Try using the scrubbie side of a kitchen sponge to clean the flapper seat in a CIRCULAR motion.

If that doesn't solve the problem get a new flapper. A red one is more resistant to chlorine in city water.

To see water going into bowl, watch for ripples with light reflections- or remove lid and see if the water level is going down before it refills.

Another thing: Fill tube gets cut to size and clipped to the overflow tube (never pushed down in there).

To prove it's the flapper... a few drops of food coloring in the tank. If the flapper is leaking then after a while you'll see the food coloring in the bowl