Trailer lights do not have to be a pain in the butt to diagnose or repair if one thinks logically about the issue. Here are some tips in that regard:
1) Isolate the problem to the trailer or the tow vehicle. Do this by disconnecting the trailer from the tow vehicle. Get out your battery charger and put a nail in the POSITIVE clip. Clip the NEGATIVE clip on the white ground terminal of the trailer plug. Power up the charger and one at a time touch the nail to each of the three remaining pins (left, right and running/tail lights). If they all work the problem is in the tow vehicle.
2) If one or more lights is out, troubleshoot that circuit starting with the bulb. If both TAIL lights are out, it may be bulbs but a problem with the brown wire is more likely the issue. If only one tail light is out, the problem is likely a bulb but the ground for that lamp may be the issue as well.
3) If none of the lights work, start with checking the ground (white) wire.
4) Each fixture must be grounded to the chassis, or a separate ground wire run from the plug to the ground connection on each fixture.
5) If the trailer uses the chassis as a ground, make sure any junctions such as the tongue on a tilt trailer has a ground braid connected around that junction so a good connection is maintained.
6) If one or more lights are out but some work fine, the fixture ground may be the issue, the bulb may be the issue, but the basic chassis or wired ground may be an issue ONLY at those fixtures.
7) As pointed out earlier in this thread, do not use the coupler and ball as a ground connection. It is simply unreliable
8) If the problem is in the tow vehicle, check the owners manual for fuse locations. If those are ok, then inspect connector wiring.
9) Lastly, if you acquire a new tow vehicle, make sure you determine if it has SEPARATE turn and stop lamps. Amber rear turn signals is a give-away here. Some vehicles may have separate lamps but still use the red lenses. Most vehicles today do have this configuration so a converter is required. If you purchased a vehicle with a tow package it will likely have the converter built-in. To correct a statement earlier in this thread, it is NOT separate STOP and TAIL lamps but rather separate STOP and TURN lamps that require the converter.