1989 Mustang license plate light malfunction: when headlights are on and push brakes, the license plate light goes out. When the headlights are off and push brakes or push trunk release inside glove box the license plate light comes on. Possible short somewhere but not sure where to look.

All lights rely on a good connection between the globe base and the metal body of the car.
This is the cause of your problem.
You'll probably find rust in the globe holder stopping the globe from making a good connection.
You may have to replace the globe holders, but it will be worth trying to clean the rust back to bright steel.

sounds like a bad ground by the tail lights

could have poor ground wire use 12 volt test light check probe socket positive wire could be grey green wires and touch ground if test light works you have bad ground to license plate lights.and license plates lights have fuse control circuit.

Sounds like either your wiring from the Light switch is bad or the Light switch itself is bad. Check with your Owners manual to see if the "Parking lights" have a separate fuse.

Your tail and License plate lights would light with either the Headlights or the "Parking" light selected at the switch as would the front parking lights. If all lights except the headlights are out, your problem is closer to the light switch than at the mounting points of each Parking light or License plate light.

This is one case where you will not know if a dead wire off of the switch is dead because of a bad switch or a bad wire. You would have to detach the harness at the switch, test the wire bundle or jump the switch with power to see how it distributes power through the Harness. It will take a lot of time to rule out the switch. It may be cheaper to replace the switch and then work from there.

I'm having the same problem on my 1995 Honda Accord.  I hit a bump hard last night, and my dash lights, tail-lights, license-plate lights, and the bell which 'bing's when the door is open and the headlights are on all stopped working.  The fuse for "Small lights" under the hood was blown.  Replaced it, turned on headlights, and it blew again instantly.
I'm guessing that I have a wire crossed (possibly inside the bulb itself) in the dash somewhere;  I checked the taillights and license-plate liights already.

Good luck!  I'll let you know how my fix goes.

It sounds like either a bad headlight switch, or a poor ground, check the body ground and make sure that the tail lights are getting a good ground. Most likely it is the headlight switch.

you do have a tail lamp relay,

could be bad headlight switch

Are the rear bulbs OK?
Is the ground good at the rear?are the sockets good?
Are there tail lights and a license plate light?then the ground is OK.
All the things that you say do not work go through the signal light switch....it may need replacing.
First make sure the bulbs and fuses are OK and power is getting to the signal light switch and to the brake light switch.
Nice car!

Probably either a bad ground (broken wire, rust) or a bad headlight switch. The brake light circuit goes from the stop switch (Located up above the brake pedal) through the headlight switch, through the light bulbs to ground. The tail light circuit goes through the headlight switch, through the bulbs and to ground. The only things that are in common are the headlight switch (Is this the switch you replaced?) and ground.
Not much, but hope this helps.
Have a super day!
Ron