Mesh fence with small openings and top rail.

I've been looking for a photo of this...

For keeping horses both young and old safely in (and roaming dogs out) I don't think anything tops this careful installation of mesh fence (note that the openings are too small for a hoof to become ensnared) with a sturdy wooden top rail.

If I were to ever lose my mind completely and breed or board broodmares and foals, this is what I would want to have. Foals can run into this and bounce off. The visibility is great, and it does not require electricity.

It does require a significant investment, but for those able to afford it, there is likely no equal in horse safety.

Metal pipe fence.

You don't see a lot of this around here. This looks like a temporary pen set-up at an event, such as a rodeo. The thin metal tubes might make movable panels weigh less, but I'd be wary of using fence that light for everyday use.

A more conventional, permanent use of metal pipe fencing features sturdier, horizontal tubes, with far less opportunities for a caught hoof or a broken metal bar.

Properly installed, metal pipe fence looks good and is one of the safer types of fence for your horse. Generally it is also an expensive option, which you may notice if you've tried to price a 60' round pen lately with 6' tubular steel panels.

Plastic/vinyl (PVC) rail fence.

Nothing looks as nice as properly installed PVC rail fence, IMHO. I wish I had it.

I don't, due to the oppressive cost of fencing four full acres, not including pasture dividers. But if money were no object, this would be my first choice.

One consideration when using this fence, is that horses can push and pop a rail out of place. To help prevent this, run a strand of electric wire along the inside of the top rail.

Another consideration is to be sure that the bottom rail is low enough that a foal (or clever pony) cannot roll under.