Start by checkin all fuses and then go through all your wirein to rear lights and the check build recepticals for wear and corosion..
It has two flasher relays. One for the Emergency Flashers and one for the Turn signals. Replace the Turn Signal Flasher Relay located front seating area, driver side, under dash, mounted in fuse panel as it is defective.
The 1987 Buick Century is equipped with a flasher on each corner of the vehicle. There will be to flashers in the front of the vehicle and to flashers in the back of the vehicle.
Well, if none of them work, (front or back) I'd suspect a fuse or the flasher. Do the 4 way emergency lights work? If so, they operate on a separate flasher and this would indicatethat the turn signal flasher is bad.
Check the bulbsCheck the sockets for corrosion or a bad ground
The flasher is located Front seating area, driver side, under dash, mounted driver side of steering column.
Faulty socket wiring problem Faulty flasher faulty switch
I'm not sure if fuses are your problem as faults would be found in several fuses to cover all the items you have listed. However, the Emergency flashers, turn signal lights are covered by fuse 7 (3rd row up, 3rd from right) in the Fuse box in the middle of the top of the dash. The instrument panel lights by 6 (1 to the left of 7). Left high beam by 27 (4th up, 2nd from right). Right High beam by 28 (to the left of 27). The other is in the 'front compartment' (engine compartment?) "STOP" by 33 (middle fuse/relay).
Replace the flasher relay, as it may have 2 relays. Front seating area, driver side, under dash, above fuse block, mounted on kick panel
If you have the four way flashers on and the flasher is not working this will cause all four lights to come on. If you are sure the four ways are off then the problem is in the signal light switch in the steering column.
Check for any blown fuses or burnt out light bulbs.
Defective Flasher Relay. Located: Front seating area, driver side, under dash, driver side of steering column, mounted in relay bracket.
There should be six signal lights on the mystique, two front, to rear, and two sidemarker lights.