Either they have it set up so the water is blocked when the filter runs out (razor-blade revenue model, which is sheer awesomeness), or you do have a solenoid valve that is blocked/frozen for the water inlet. The 'hum' you hear is the power supply running juice through the valve, so that means you do have 12 or 24V power going to it (if you have a voltmeter, you can check that on the contacts, too). There are 2 of these solenoid valves on your water inlet part - one to feed the icemaker and one for the water dispenser.
I have had those valve go bad on a refrigerator, and a top load washer back when I was in CA.
But generally, when the solenoid valves start going bad, the water dispenser will drip water before it totally freezes up (valve doesn't close all the way); your problem sounds like it is frozen shut, and if you have hard water, that could be the issue.
If you know your refrigerator model number, you can look up the part here:
http://www.repairclinic.com/
This is what they typically look like:
They are very accessible on the lower back panel of the fridge (the water line goes straight into these). BUT, make sure you have a valve shutoff for the water line to the fridge before you do anything - many people to 'self install' ice/water capable fridges don't add those in when they tap off a sink line, which makes it a giant pain in the ass to work on them, as you have to shut water off at the sink or the whole house.
If you can identify the solenoid valve part on that site, it will only require a screwdriver and wrench (plus maybe some Teflon tape) to remove the old part and pop in the new one. It is just 2 spade-lug electrical contacts (low voltage) and water connections no more complicated than a toilet or sink. You could also take the old part out and see if you can wash/clean out the valve components, but that's a temporary fix because once the hard water buildup starts, unless you chemically clean it, it will clog up faster next time.
You can buy the part from this site, or you can find a local supplier - I used to buy mine locally once I looked up the P/N info here, so if I found out it didn't fit or wasn't correct, it was easier to return.
Most of these types of things on your household appliances are 1950's-Era technologies, and are not complicated at all to do yourself (can probably find Youtube videos that walk you through them as well). If you call someone else to do it, expect at least $50-75 in 'site-service' fees just for showing up, plus hourly charges plus parts, which will generally cost $150-250 total vs. the $25-50 if you buy the parts and DIY.
