Reverse lights not working.

You need a test light or voltmeter-you cannot do electrical tests without one or the other. Where you start is at the brake light fuse-pull it out and check it. Also use the test light (or voltmeter) to verify that there is power at the fuse-one of the fuse terminals should have power to it, it is hot at all times so anytime battery is hooked up, the brake fuse has power to it.
Next you go to the brake pedal where the brake light switch is. It has a button, when pedal is depressed, the button pops out and power goes through the switch and on to the next...
If you have a high mount brake light, look at it, it should be lit when pedal is down, there is a wire direct from the brake pedal switch to the high mount stop light.
Now on most American made cars, the brake lights and the turn signal bulbs work together, so the brake light wire for the rear bulbs first goes from the pedal switch to the turn signal switch before routing out to the rear brake lights.
Use the test light to verify power is at the brake light switch on the brake pedal, when switch is active, power should be on both sides of the switch.
Next you would check for brake light "power in" at the turn signal switch. Keep following the power path, eventually you will arrive at the rear bulbs. If you pull the bulbs out, you can use the test light to check if power is there at the bulb socket-when the brake pedal is depressed. If these are two filament bulbs and sockets (one filament is for the tail lights, and the other filament is for the combined brake and turn signal filament) you have to be sure you are testing the right socket terminal for the brakes/turns filament.

Buy a cheap test light. Practice with the lead clipped to the negative battery post or terminal and use the tester probe to touch the positive, tester should light up, see how easy? Now when you are away from the battery and looking for a good place to ground your tester, clean off the frame, or clip onto a metal bolt, you have to keep testing until you are sure you are connected to a good ground. If you are under the dash, clip lead to a dash frame or bolt, then touch some of the fuses little ports or slits on top of fuse that are open to the fuse metal inside, to watch the tester light up- the key may need to be in ON, not start. When the tester lights, then you know you have a good ground. Now you can continue with testing.

No they should only come on when you put in reverse or else everyone would think you were backing up! its your bulb if just one, the fuse if both or the reverse switch indicator... Good Luck

if the bulbs are good, then check the signal from the transmission range switch.you can check that at the bulb socket with the park brake set, the key on/engine off and put trans in reverse. use a test light and check power in the light socket. no power indicates bad fuse, bad wiring, or bad range switch.

check circuit board at lights and grounds, it is best to use a Power Probe to test as you can apply power or ground at the lights to verify problem.

Im not sure the make ,year or model of your car but , the next thing to check as you already checked the fuses and brake light switch would be to check the wires going into the socket, the sockets go bad often so using a test light the next thing is to check the power and ground right before the socket, check the wires at the end before they enter the socket, if you have power and ground then the sockets are bad. you also know that the brake lights and the third light use diffrent power sorces, so just because the third light works doesnt mean the fuse is good for the other brake lights,good day,

check to make sure that you have the correct bulb in the correct socket in both left and right tail lights. dual filament bulbs have to contacts on the base of the bottom of the bulb. make sure the socket that you are inserting them into also has two contacts. if that doesn't work then you need to get a test light so you can find out if power is going to the bulb housing and all ground. check all fuses also with amp meter

If you have tested fuses and bulbs with test light,and they check out,
the next step is to put you jeep in reverse with the key on,and use your test light to probe the wires before they enter the bulb sockets for power,if you have power replace sockets,if not check to make sure wires are still connected to back up light switch on trans,if you find no problem there,replace the switch

first check the bulbs, then the fuse. if it is not either of those then it might be the switch that is activated when you shift to reverse. this is usually a plug on the side of the transmission. there will just be two pins in this plug. to test and make sure that everything else is working you can unplug this plug and use something like a paper clip to make a connection between these two pins on the side of the plug that you unplug from the transmission, not the transmission side. if connecting those two pins together makes the light come on then the problem is inside the transmission and it would have to be taken apart to fix it.
hope that helps

Questions? Do any of your brake lights work? Including Center Stop? Do your parking/taillights lights work? All your bulbs are multi-filiment bulbs that work through mulitple sources. Knowing which bulbs are doing what under all conditions might help narrow down the problem. Meanwhile, check for continuity between, stoplight switch and multifuntion switch (white wire). Also, check DrkBlue/White from MFSw. to Boby Control module. BE SURE to use a low inpedance tester to protect your modules. The DrkBl/Wh wire is your rigth side input. I may be a good idea while testing these wires to wiggle them to look for intermitent shorts.