The short answer
Two habits prevent most problems: check the tyres and the wheel bearings at every stop on a long trip, and inspect the seals and the roof once a season. Everything else on this list matters, but these two account for the majority of roadside stops.
Tyres: age, not tread
Trailer and RV tyres are of a specialised construction with high pressure and thin sidewalls, and they age from heat and load rather than from wear. Cracks in the sidewall, bulges, or more than four years old with visible ageing should drive a replacement decision, and most manufacturers and trade bodies advise replacing serviceable tyres after roughly six to ten years regardless of tread. Check cold pressures with a proper gauge before travelling, because a hot reading taken after a long run reads high and will mislead you. Under-inflation is the single biggest cause of trailer tyre destruction, and it does not show up from outside.
Bearings and brakes
Wheel bearings should be inspected and repacked on a schedule set by the bearing manufacturer, typically once a year or every ten to twelve thousand kilometres, whichever comes first for a towed axle. Listen for a constant drone that rises and falls with speed, feel for heat on the hub after a long run, and check for play with the wheel off. Brake pads and shoes on towables wear with the load, and electric brake magnets need their adjustment and their wiring checked annually. A trailer with a seized bearing at a roadside is a very expensive afternoon.
Seals, roof and awning
The roof is the most frequent source of real damage, and the failure is almost always at a seal: around the vents, the skylight, the air conditioning unit and any roof penetration. Inspect it dry, from a ladder, in daylight, at least once a season, and re-seal with the product the manufacturer specifies, because sealants are not interchangeable. Slide-out wipes and bulb seals harden and tear, so check them after any long travelled period and before a season in storage. An awning left wet for days is a mildew problem; one caught in a strong wind is a replacement.
Systems worth a yearly look
Battery condition and terminal corrosion, the water pump and its filter, the toilet seal, the gas cylinder and any flexible gas hose, the carbon monoxide and smoke alarms with their test dates, and the spare wheel. Test the alarms by pressing the test button rather than assuming. A five-minute annual check of the gas fittings with a leak detector will prevent the worst class of RV incident.
Before every trip
Tyre pressures and condition, lights including the trailer's, the coupling and safety chains, the breakaway mechanism on a towable, the jack and the wheel wedges, mirrors, and the internal checklist you actually use: keys, documents, water, level, and the handbrake. Write your own short version and keep it by the door, because the items you forget are the same ones every time.