it is my grandmothers car and she has been stuck with it not starting up a couple of times. It has been in and out of the shop for the same problem. We had the starter changed about a year ago, that didn't correct the problem. The mechanic says he has contacted about 9 other mechanics and have all stated various things that could be the problem. The alternator seems to be okay, the battery is fine, and apparently the fuel inlet valve has been changed. Regardless, the car will still randomly not start up.
she describes the noise it makes as a humming when it won't startup. No grinding, so I ruled out, I haven't been able to duplicate the problem. Any help or suggestions would be greatly appreciated. Most recently we had the connections to the starter tightened, but issue persists.

Does it crank but not start or does the starter motor not crank?

If the starter motor doesn't crank, she may be hearing the fuel pump in the gas tank run (hum) for several seconds, as it pressurizes the fuel rail.

I would begin by removing and fully charging, or replacing, the battery. Most auto parts stores will test and charge batteries for free. Mercury Sables are extremely reliable cars. I doubt that anything major is wrong. Make sure she has changed the fuel filter in the last 30,000 miles and that the spark plugs have been changed in the last 3 years. Only use the same Motorcraft platinum spark plugs that came in the car.

At this point, my bet is that the main problem is with the battery. Don't assume it's O.K. just because the lights and horn work. It may be that the condition of the spark plugs and fuel pump have caused her to drain the battery, as she tries to get it started.

Persistent humming, or only when the key is in the start position?

At first I thought it may have been a clogged filter or something, but you seem to be describing problem more related to the ignition and electrical system rather than a fuel problem.

I believe on your car you have a separate location for the starter and solenoid, maybe you are having issues with that if it isn't the starter itself. You can check by directly crossing the power wires in it. It would be the two thicker ones, not the smaller ones. One comes from the battery, and then the other goes to the starter itself. Also check for bad grounds.

Good Luck.

Thanks guys. Appreciate your knowledge and expertise in the matter.
Txgreasemonkey: the starter motor does not crank. So that could be indicative of a fuel problem?

Infojunkie4life: the humming persists the entire time it is in the startup position. I will mention the solenoid to the mechanic and see what he has to say about it. Thank you very much.

Oh didn't see the updated post. Okay. So the battery has been replaced but I am unsure of the spark plugs. The fuel filter has also been replaced within 30k. Okay, you guys have given me some things to look into, so I greatly appreciate it!

Since the starter motor does not crank, this link should help:

https://www.askmehelpdesk.com/cars-trucks/faq-how-troubleshoot-repair-maintain-hondas-selected-other-vehicles-46563-3.html#post235560

Thanks for all your assistance. Put a new starter in, running smooth now. Thanks all.