If you've read my first two posts on breathing, you've probably noticed something missing in the connection between martial arts and proper breathing. Relaxation gets mentioned repeatedly - yet total relaxation is not exactly helpful in the martial arts. Getting hit in the stomach with relaxed abs is not so much fun - as anyone who's had it happen to them can tell you.
So once you have some idea of what good, relaxed abdominal breathing feels like, the next thing to work on is appropriate tension.
Tension serves several purposes. Supportive tension helps us to move most efficiently - like the tension one uses to stay in good posture - without the support the skeleton slumps, and movement (and breathing) becomes more difficult. Supportive tension is described well here - and the Alexander Technique, as well as yoga, Pilates, or the Feldenkrais system can all help to teach good general body mechanics. My go to book for working on my own posture is an odd little book called The Vance Stance. There are plenty of other posture and body mechanics books out there, but this one seems to be particularly helpful from a self-help standpoint.
The other major tension, and the one more specific to the martial arts is protective tension. Protective tension is why you tense your abdomen before a blow to the belly. You're protecting the vulnerable organs underneath by hardening the surface above them.
The tricky part is allowing for functional tension without interfering with the proper flow of breath. This is where you want to go back to the wall. Take a front stance with your arms slightly bent, and really push into the wall. Push it like you want to push it down! Yes - this is the same drill to use for making sure you're not breathing with your chest muscles. This time you want to pay attention more precisely to what your abdominal muscles are doing. It's possible to push into the wall with just your chest muscles, but for a really hard push you're going to be engaging your abs too - what you're watching for is how it feels to tighten your abs and still breathe by moving your abdomen in and out. To me it feels like I'm holding the very front sheet of muscles taut while deeper muscles move that sheet in and out - but that's not how it feels for everyone. Play around with it and see how it feels for you.
In singing this abdominal tension is called breath support, and it allows for precise breath control both in and out. One place where singing support differs from what you want in a martial arts setting is that in singing using every last dreg of air is not particularly avoided. In fact deliberately getting rid of all your air so you can pull in an entirely fresh lungful is common. After all, you may have to stretch that lungful out over a very long phrase, so you want as much oxygen as possible. In martial arts - particularly in sparring - you never want to be completely out of air. Blow out to take air in, yes - that avoids breath holding - but only blow out about 60-80% of your air. Two reasons for this: 1) In sparring, you can't control your opponent, so you never want to be locked into having to breathe right now or else. That might be the very moment your opponent attacks. 2) When your air is blown completely out, your organs are maximally compressed and your abdominal wall is sitting right against them. There's no cushioning for a blow except the strength of the abdominal wall itself. Contrariwise, a slight blowing out of air (under tension) while being hit can function a little like a floor slap in a fall - it takes up some of the force of the blow.
The other major difference is that in singing, dropping belly tension entirely in order to suck in breath as quickly as possible is downright common. You may have only a split second in between phrases to get in a bellyfull, so you need to be quick about it. This is obviously a bad idea in a sparring situation. No abdominal tension is a big vulnerability in sparring, no matter how brief, nor does kumite have the kind of pre-planned breaks that you can build into even a rapid song. In kata it is possible to grab breath this way, but I think it's a bad idea to establish the habit of ever dropping protective tension. Better to build in the exhales and work on learning a rapid but controlled intake. Any situation where your abs are engaged is a good opportunity to play around with breath control - see what it feels like to inhale at various places in your push-ups (you are holding your core during push-ups, right?), or during crunches. The more you know about what good breath control feels like for you, the better you will be at maintaining it in sparring when a thousand other things are happening.