First of all, I don't necessarily think I'm qualified to attempt to answer this question. I've been riding for a number of years, but that, as we all know, doesn't mean I know much, or that what I think I know is right. I'm pretty sure that I know more now than I used to before I started on the journey I'm now on (see the sidebar "Steps on the Journey" for more on this) - to work with horses in a more thoughtful, respectful and effective way - but I've got more to learn and probably always will - and the horses are my best teachers - Dawn is my current teacher and she has lots more to teach me yet.
The first answer to the question isn't about technique - what aids to use or what to do. There are lots of technical things - training steps and exercises - which will help develop softness - but there are some more basic things that have to be there first. Fundamentally, the answer to the question is: work on yourself and a lot of the rest will fall into place. Almost any horse is capable of carrying itself softly without a rider in the pasture - things get messed up when the rider enters the picture, and sometimes we're dealing with issues that have been trained into the horse by prior riders. If we can fix ourselves, the horse will meet us more than halfway. For example, in the last photo from my picture post about Dawn, I am doing exactly nothing other than offer her a feel and some mental direction. That's right, nothing that's technique-based. I'm not putting on any aids, or pushing, or pulling, or doing anything except be "in" her as she softly carries herself around a turn. I'm just with her, and we're doing the work together.
Now, what does all this mumbo-jumbo mean? I'm firmly convinced that any horseperson, no matter their level of experience or training, can make progress down the road. Here's a couple of things that have helped me over the years - the following posts have some more specific "technique" ideas to try:
1. Be soft yourself. This is a lifetime project - work on being soft yourself, in mind and body; being in balance, physically and mentally. This isn't about horses, it's about life and how you live it - but then horsemanship is life and vice versa. When you walk, walk softly. When you pick up a plate and put it down, do it softly. When you close a door, do it softly. It's a lifetime project for me. And take your time - part of softness is letting things take the time they need. If you're in a hurry, things will come more slowly than you want - and remember, horses don't wear watches.
2. Listen to what the horse is telling you, or for that matter, what anyone is telling you, even if you don't like what you're hearing. When you ask the horse for something, really listen to the response - if the horse is saying that you're unclear, or inconsistent, or abrupt, or unbalanced, then fix it - that's your job.
3. If your intent is good, don't be afraid of making mistakes. If you bring good intent to your horse, and try something and it doesn't work or the horse gets worried, your horse will forgive you. If you want your horse to forgive you for a mistake, forgive yourself first. Give yourself and the horse room to try things, whether they work or not - that's how you figure things out together. Don't miss the good by chasing the perfect - things have to be worked out incrementally, and sometimes things go backwards before they go forwards. Sometimes the horse struggles with something, and as long as you're there to help them get through it, that's OK - don't leave the horse struggling.
4. You have to know exactly what you want the horse to do, ask, then find the try, and give a release. When your horse answers your question with a try, notice the slightest move in the right direction and reward it. It's these tiny steps that build confidence and develop a conversation that leads to the result you want. It won't come all at once and it takes time - but it's the releases that build the confidence and shape the learning.
5. Find good teachers - people and horses. With trainers and teachers, follow your gut and trust your instincts - if it feels right, it likely is, and if it feels wrong, run for the hills. Every time I've ignored my gut instinct, I've regretted it. If your teacher doesn't follow principles that make you comfortable, you've got the wrong teacher, although sometimes even a bad teacher has something to teach you. Just make sure your horse isn't harmed when you're with the wrong teacher or trainer. A teacher/trainer who isn't open to new ideas, thinks there is only one way to do things or feels threatened by other ideas, isn't a good trainer. A teacher/trainer who won't answer your questions about why they're doing something, or asking you to do something, isn't a good trainer. Make sure the trainer/teacher you work with teaches you how to work with your horse yourself and doesn't just train the horse for you. Figure out what every horse you ride has to teach you and learn from them - the horses are the best teachers. If you get a chance to audit or ride with someone really good like Harry Whitney or Mark Rashid, or the good people who exist in every discipline, do it, and do it again, until the lessons and examples begin to sink in. Watch people work with horses, in person, and on videos and TV, and learn to make your own judgments about what's right and what isn't. Read books, and more books. Learn and learn and learn some more.
6. Find the good and build on it. Ignore the stuff that doesn't go right - and lots will go wrong. Notice what goes right, even when a lot is going wrong, and build on and reinforce what is going right - every step on this path takes you towards your goal. Focus your energy on what you want, not on what you don't want.
7. Practice, and practice and practice some more - and do what you can with your horse. No one said you have to be an expert to make progress down the road. Learn to feel what is happening, in your own body and that of the horse - most of the important stuff is about feel, not technique. Take responsibility for and ownership of what you do with your horse.
I expect this doesn't help too much from a practical point of view. The techniques I use to help my horses be soft derive from all the stuff above. It really helps when you're getting started to have a knowledgeable set of eyes on the ground so you can learn what things feel like, so you don't miss the tries and learn to give immediate releases - it's this combination of feel and timing that can be the tricky part at the beginning. The most common mistake is to release too late - err on the side of releasing too early, just as the try starts - but don't release on a brace. And read both "Common Themes" posts from the 2009 and 2010 Mark Rashid clinics, on the sidebar - there's a lot of good ideas in there that have helped me.
The following posts will have a few simple exercises you can do to begin to get the feel of softness. Some of them don't even require a horse - a friend will do the trick!