The simple answer to this question is ANY! As long as the saddle fits the horse and the horse is already well schooled in basic equitation for the discipline you wish to ride - Hunt, Saddleseat, Western etc. any breed of horse can be used to ride aside!
Most of us envision an athletic Thoroughbred..
...or maybe an beautiful Arabian.
I dream of riding aside on a Saddlebred with their long elegant upright neck!
..but what about other breeds? Here is a Morgan...
Here is a Pasa Fino..
...or how about a Mule???