Horses are very curious animals and so it is the responsibility of the owner to make sure that these animals are provided with the safest type of fencing to fit the scenario in which they are kept. Often times different types of materials are used for corrals and exercise pens as opposed to pasture and turnout.
Almost everyone that owns horses will agree that the most dangerous horse fencing is barbed wire. They can get tangled in the barbed wire and injure themselves horribly, sometimes leaving life-long scars. They will stick their necks and legs/feet through the strands to get to hay, or grass, on the other side. The other concern is visibility. If they do not see the fence it is possible that they can run into or through it.
Ranchers and landowners with many acres of land might decide to do cross-fencing on the property. This allows for horses to be rotated within the property, so that there is no overgrazing in a specific area. There may be many pasture areas designed to fit the need for grazing.
Basically then there is perimeter and cross-fencing. The latter type can also be portable, not permanently installed, which allows for fairly easy relocation. Perimeter can be made from many materials such as wood, smooth wire, barbed (not recommended) wire, braided cord, webbing/tape (usually has stainless steel woven into it), synthetic wood, electrical, and pipe. With all types there will be maintenance required to keep it in good and safe working order.
Expense is another consideration in determining which type of material to use. Wood is very expensive and does require a lot of maintenance. This is a very classic type and can be very safe as well. If a horse does collide with it, it will break away. There is the possibility of splintering wood and exposed nails which could injure or impale. Using a synthetic wood type of material can often times be even more expensive but does require less maintenance and might be a safer alternative to wood.
Wire is very affordable and low maintenance. About the only thing that needs to be done on a regular basis is to make sure that the wire strands are kept tight so the horse does not get tangled up in it. Elect to run a visibility tape on the top row of wire for safety, and if necessary this type can be electrified. This will ensure that they do not rub up against or poke their head through the strands. Be sure to hang signs that state the fence is electrified, or hot, so that people do not touch.
Pipe is a very common type of material used for corrals and exercise pens. Welded in panels they can be configured into many different shapes and sizes. This is another virtually maintenance free material that is fairly easy to install.