We have a banging (water) pipe / vibration noise problem that just started about 3 days ago when we flush the toilet in our master bathroom.

It appears to occur several seconds after flushing... when the water tank is finished refilling and the water shuts off.

Although we've isolated this problem to one toilet (other faucets, showers, tubs, and toilets don't cause the noise), a similar but less vibrating sound occurs when we run the washing machine or use the water dispenser on our refrigerator.

I thought I'd start troubleshooting with the toilet, since its noise is the most pronounced. And if it's helpful, our toilet valve/ball mechanism is a FluidMaster 400A.

We have several water pipes that I can see in the ceiling of our basement, water lines that go to other parts of the house, but I think the troublesome toilet, washing machine and refrigerator are fed by the same line... since all three of them are on the same side of the house in proximity to each other (1st floor: washing machine and refrigerator. 2nd floor: master bedroom toilet).

I can't think of anything that we've done differently or unique before the noise started a few days ago.

Any suggestions would be greatly appreciated.

Sounds like water hammer to me. Water hammer is usually recognized by a banging or thumping in water lines. The noise occurs when the flow of moving water is instantaneously stopped by a closing valve. This sudden stop results in a pressure spike behind the valve which acts like a tiny explosion inside the pipe. This pressure spike will reverberate throughout the plumbing system, rattling and shaking pipes, until it is absorbed.
Back in the 60's we were required to to put air chambers on all bath lavatories and kitchen sinks. Over time water and condensate built up in them and they must be recharged. First turn off the house at the main house shut off valve. Then open hot and cold faucets at the farthest bathroom to prevent air lock.(This is important.) You will find under each fixture little brass or chrome valves. These are called angle stops. With a small pail held under the angle stop to catch the run off look under the handle of the stop and you will see a nut with the stem running through it. This is called the bonnet packing nut. Put your wrench on it and back it off counterclockwise. Now open the stop and remove the stem and washer assembly, The stop will began to drain. When it has quit draining reassemble the stop and move on to the next one. Do this on each one of your angle stops. There will be six of them in a two bath home. When you are done, close off the faucets you have opened, turn on the water to the house, and give yourself a pat on the back.
You have just recharged your air chambers and saved yourself a hefty service call bill. Hope this helps you out and thank you for rating my reply. TOM