You can always rely on Robert to be rude.
The hideous consequences of semi-trailer and caravan jack-knifing has caused vast amounts of research to be carried out. Unsurprisingly there are lots of factors involved, but it is a fact that a short trailer with a low centre of gravity and the load centred above the axle(s) is more stable than any other combination. The longer the trailer and the further the load from the axles the worse the stability because of the pendulum effect. So the basic design of the IWT horse trailers is very stable.
The double axle adds one advantage and one problem. The advantage is that that when turning the axles follow slightly different radii which adds yaw resistance from the tyre walls and hence stability: anyone who has tried to manoeuvre one by hand will know this. The problem is in the cart spring suspension which means that on uneven surfaces all the load can be taken on one axle while the other one lifts off the ground. This would cause overloading of the tyres except when virtually unladen
Nose weight is another issue which can be difficult but I've never seen anyone measure it on a horse yard in the past couple of decades. As Robert notes too little is by far the worst. The generally recommended weight is 7% of the trailer weight subject to the restrictions on the towing vehicle. My horses tend to stay put once loaded into the vehicle, but we always use the centre partition.
I've never seen a horse trailer with a stabiliser and rarely seen a caravan without, and round here horse trailers, particularly Ifor Williams trailers, outnumber caravans.