This article is from the Fishing bass FAQ, by Bass Rogue with numerous contributions by others.
The most important part of troubleshooting a trailer problem is a good
visual inspection. Check all ground connections and make sure they
are clean, mechanical sound connections that are protected from the
elements. Examine all bulbs and light fixtures up close. Look for
water trapped inside the light fixtures. Look for discolored bulbs.
When in doubt, fix the connection or replace the bulb or fixture.
One of the most common problems with trailer wiring is a broken ground
connection. If you are troubleshooting a trailer, do not connect the
trailer hitch to the tow vehicle ball and do not connect the safety
chains or cables. These can provide false grounds to the trailer.
The only connection between the two vehicles should be the four-pin
jack and plug connection. The way I prefer to troubleshoot trailer
wiring problems is with an inexpensive 4-amp battery charger with a
current meter on it. Do not use one of the more expensive chargers.
Some of them have protect circuits that won't allow any current to
flow unless they detect a battery on the end of the charger cables.
The first test should be one of the turn/brake lights circuits. These
are the simplest circuits, usually only one bulb filament is involved,
sometimes two. If you are using a battery charger, connect the
negative cable to the white wire pin (ground) on the trailer plug and
connect the positive battery charger lead to the yellow wire pin. If
you are using a tow vehicle, turn the left turn signal on. The left
turn light should come on. If it doesn't, first check the bulb, then
the wiring to it. If one of the lights on the other side comes on,
see the Strange Trailer Problems discussion below. After checking the
left turn light, check the right one. Leave the battery charger on
the white wire pin and move the other lead to the green wire pin; or
turn the tow vehicle's right turn signal on. The right turn light
should come on. When you test the turn signals, you also check the
brake lights. They are the same circuits.
The last check should be the running lights. Leave the battery
charger on the white wire pin and move the other lead to the brown
wire pin; or turn the tow vehicle's right turn signal off and turn on
the parking lights. All the trailer running lights should be on. If
not, check the bulbs, then the ground connections, then the wiring.