In the last week or so, several times a day in my home, a 60-year-old two-storey ranch-style house, I hear a humming sound as if the plumbing pipes were vibrating. The noise can be quite loud but disappears after a few seconds -- ten at most. My plumber thinks it is water hammer, but it is not a single loud noise; it's an ongoing hum. Should I worry?

Well, that does not sound like what is generally referred to a water hammer. Water hammer refers to a loud bang when a fast acting valve (like a toilet) closes.
Humming sounds are generally a loose washer in a valve somewhere.

Since this occurs periodically through out the day and last for about 10 seconds I suspect that it is in the pipe leading to the ice maker of your fridge. Ice maker valves are generally open for 7 seconds to refill the tray.

I have a different take on this. I've had the same problem with toilets.
Let's reverse the washer first. Shut the water off under the tank and take the lid off. Remove the three screws on top of the ballcock,(sae image) and pull the stem,washer and float assembly up. Pry the washer out and turn it over. Seat it back firmly and see if that doesn't help. Before you reassemble hold your hand over the ballcock and turn the water back on to flush out the supply.
Sometimes the problem is caused by the ballcock trying to fill while water is seeping out of the tank. The sound you hear is the washer vibrating against the seat of the ballcock. First check there linkage between the flapper and the flush lever. It should have 1/4" of play when your tank is filled. The water level should be 1/2" below the white overflow tube after the tank's filled. Next feel the flat side of your flapper. Does it have wavey edges? Do your hands come away black? Replace it if it does. And last check the seat itself. Any nicks or rough spots? If so click on back. There are replacement seats that glue right over the old one and are easy to replace. One of the above should take care of your howling.

This happened in a new house some years ago. Only at night and in the early morning. They thought the house was haunted and were about to move out. I caught it in the powder room toilet. It would vibrate and the pipes in the wall would pick it up and carry it all over the house.
One more thing. This could also happen if the ballcock kicking on every time you shut off a faucet. To check this out remove the lid and open and shut the shower valve real fast while watching the float ball. If it bobs up and down you have located the source of the problem. It is a sort of water hammer and you don't recharge your air chambers by simply draining the system. It's more complicated then that.
Let me explain. 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 ratting my answer. TOM

Yep, Tom's the man, hadn't thought about a leaking flapper and a loose washer in the toilet. See what experience will do for you.