1998 lincoln TC brake light won't work

You hear a clicking sound but none os the turn signals work, neither front, rear or dash indicators? Check the underdash fuse panel to see if it is the flasher that is clicking. There needs to be some resistance in the flasher to make it work so some light should be working. Do the emergency flashers work? They use the same bulbs but a different flasher. Both systems go through the turn signal switch, Move the emergency flasher lever in and out several times to make sure there is a good connection. Check to see if current 12 volts are getting to the signal bulbs.

Has the vehicle been in an accident on the left rear side. First check the bulb to be sure it is not defective, trade it out with the side that works properly to test. If the bulb works well then My thought are you may have a shorted wire. The turn signal flasher works by heat build up within the flasher the heat is caused by the load of the bulbs, the load heats the metal spring causing it to open the circuit then as quickly as it cools down the circuit closes again lighting the bulbs.
There is also a possibility of a defect in the turn signal mechanism on the steering column. The turn signal switch on many models does three different things on many models that use the same bulb element / or bulb for brake and turn on rear of vehicle. It controls the circuit for brake light by sending power both left and right side rear when turn signal is in neutral position then when turn signal is in left turn the switch cancels brake light power to left side rear and supplies flasher power to left side while maintaining brake light supply to the right side rear, and so on opposite when the right turn is chosen on turn signal switch. The turn signal indicators on the dash normally run off the same circuit as front turn signal bulb only which is also controlled by turn signal switch. Front has turn signals but no brake light.
I think there is an additional load on the left side rear turn circuit and it could be either a wire that is shorted or the turn signal switch if the same bulb element is used for both functions.
Often times if trailer lighting is connected the flasher will also blink a little faster because of the extra load of additional bulbs.

indicates a problem in the harness to the rear harness
cut wires , incorrect additions to harness
possibly along the chassis or under body

Most likely this is a brake light switch issue. Replace switch.

If your emergency flashers don't work, all flasher bulbs are fine, but flashing too fast, then check and replace Hazard light switch.

There is a separate switch for the third brake light.
Put on your flashers to ensure that your brake lights are working.
Then first look on your gas pedal linkage to locate the brake light switch.
Gary

replace flasher fuse and or check to see if middle brake light is blown

It appears to be the mutifunction switch. (Turn signal control). That, plus the 3 words in parenthesis wil come to 699.00 bucks. then we add the switch.... Just kidding. The multi finction switch is defective and about to fail altogether.. Sorry.

Hazard light switch... They all intermingle in operation.
My ex wifes 96TC was doing the same thing... I could OCCASIONALLY get the hazards to work by turning the hazard switch on/off like 10 times in a row and suddenly it would fix... finally bought a new switch

If only your brake light in middle of rear window works then your other bulbs are bad. If brake lights all work in the back than bulbs are good. If you put turn signal on right or left and the bulbs light in front and back then you need to change multifunction light control box under dash. My understanding is it cost $600 new. So I would try used.

try your flasher relay in fuse box