Front right signal lights come on but don't blink, rear right does not come on at all. Left side is working fine.

Check front signal bulbs.

Generally speaking, symptoms of a bad or failing turn signal bulb are usually the culprit...in your case, either it's the front left or rear bulb. Varies for different vehicle makes...common signs include:

1. turn signal light staying on solid
2. or blinking very fast
3. or either sides of the bulbs stop blinking altogether.

To test this, make sure you reconnect everything you disconnected.
Turn the left signal light on; go to the front of the vehicle, is the turn signal bulb illuminated? If it lights up, even though it is not blinking obviously the bulb is okay. If it doesn't, replace it- problem solved. You'll do the same procedure for the rear turn signal Is that lit up? If not replace the bulb and the problem is solved.
Another common failure is when the turn signal blinks really fast, same procedure as mentioned above.
If both bulbs for the left side and right side illuminate but don't blink, the problem is in the steering column turn signal switch.
If the left or right side blinkers don't work properly at all...then check the integrated hazard warning/turn signal flasher relay- that may be defective; again, that's usually the case if both sides of the signal light don't work.
If you are blowing any fuses related to the flasher system, you have an electrical overload in the turn signal circuitry.

If the rear turn signal lights are out, and the front lights are ok but don't blink then, the bad bulbs in the rear turn signals should need to be replaced.
Or there could be a bad ground connection.
Also, there may be two relays for front and back lights.
Hope this helps.

It seems to be a ground defect on right side. Do you have the same bulb's number . What about the front flasher. remove right rear lamp assembly remove connector measure voltage between ground and current .reconnect it measure voltage at bulb socket. you should find your problems
hope it helps

bad lamp is my guess,

turn on park lights and signal on right side and take a good look at the front and rear light , see if one is dimmer than the other this may be due to a bad ground and when the park lights are on and drawing more power it also needs better ground, check the sockets front and rear try wiggling the bulb to see which one is causing the problem

YOU HAVE A BAD GROUND FOR FLASHERS OR FRONT YTURN IS OUT AND RUNNING LITES HAVE FUSE GOOD LUCK LOOK AROUND FUEL TANK FOR TORN OR PINCHED WIRES

Replace the turn signal switch. Bad contact inside causing a crossing of the circuits. When the turn signal is applied it should open the brake light circuit allowing for just the signal to work. Doesn't sound like the switch is doing it's job.

make sure our lights at rear are grounded sounds like a bad earth

Sounds like a ground problem on one or more of the light modules.