I've got a couple of kid's trailers. Not sure of the brand. One has wheels attached on one side, the other has a hoop around the trailer and wheels attached on both sides.
It seems as if the one sided wheel connection is a bit loose, especially with heavier loads. The two sided connection works well. The fabric on the trailer is wearing out though. I took the superstructure off of one of the trailers, but worry about no fenders.
Both of those trailers have a clamp onto the chainstay, and a spring for flex. They have a problem with surging when loaded with 100+ pounds.
My favorite trailer is my homebuilt. Aluminum frame. Straight-pull hitch based on a Heim joint (which doesn't quite have enough flex for the needs, but would be better if I actually put a kickstand on the bike). Wheels are rear wheels from 16" Schwinn Stingray bikes. Unfortunately I was backed up to a deadline when it was built, and it never quite got finished, but it still is a strong trailer, capable of hauling more than I can pull (500+ lbs).
I have a covered one in the planning stages, but not complete (see the load on the trailer).
The design was to have a second row of frame for fender protection and two tailgates which would fold flat to make a flatbed.