When discussing "flexibility" in the horse, I naturally lean toward exercises that build strength and bending ability in the poll, neck, shoulders and spine. While most disciplines tend to have their own specific exercises and movements created to make a horse more supple and willing to bend, I take a more basic approach that I feel is beneficial to all riders, horses and disciplines.
In order for a horse to be flexible, he must first be balanced, and balance is something that is achieved through more advanced training. Still, whatever your riding level and your horses training level, it is important to start with a good "warm-up" that loosens your horse's neck, poll and shoulders. Begin walking your horse on a loose rein around the perimeter of your arena, each time that you arrive at a "corner", ask your horse to bend (by applying inside leg pressure and direct inside rein pressure, in effect asking your horse to bend around your inside leg) in a circle; start with a ten foot circle and slowly spiral down until you have made three circles of consistently smaller size and then ask your horse to continue on around the arena until you reach the next corner. I call this warm-up exercise "Corner Circles".
Next you will ask you horse to enter the center of the arena and begin walking a circle (not too small at first), slowly use outside rein pressure and outside leg pressure to achieve a "counter-arc" so that your horse is crossing over in the front and walking a circle with his head tilted to the outside. This warm-up is called the "Counter Circle", and will help to achieve suppleness and flexibility in your horses spine and shoulders.
Now that your horse is properly warmed up and loosened up, you can move on to an exercise that will promote overall balance and flexibility. This exercise has been made popular by Dressage riders, as it is one of the staples of training for that sport, but it is just as commonly used by riders of Western Performance horses, Endurance riders, and even show competitors; this exercise is most commonly called the "Spiral Circle Exercise". What you will want to do is clear an area in the center of your arena and place an object there for reference, you will then begin by walking a circle around this object approximately 50 to 60 feet in diameter. Each time that you complete one full circle, ask your horse to flex inward by applying aids (inside leg to achieve the proper "arc" and inside rein pressure in addition to the steadying pressure of the outside rein and leg) thus making a smaller circle. The smaller that the circle becomes, the more difficult it will be for your horse to maintain the proper arc, this is what builds muscle and suppleness. Once your horse completes one full circle as small as he can, begin spiraling back out. You should start this exercise at the walk and eventually move on to the trot and the canter.
In the end, flexibility cannot be achieved without balance and softness, from both horse and rider. If you are rigid, your horse will be rigid. If you begin with the proper warm-up and work on the above exercise, regardless if you ride English, Western or even simply enjoy being on the trail, you will find that as time passes your horse will be more comfortable and willing, in turn making your ride more fulfilling.