The only specific type of breathing I know is belly breathing, or diaphragmatic. Basically, your diaphragm (and not your chest) should be doing most of the rising and falling to breathe. As a long time asthma sufferer, this form of breathing has been brought up to me since I was a child. A 1992 study states:
Deep diaphragmatic training resulted in significant reductions in
medication use and in the intensity of asthmatic symptoms.
Importantly, a nearly 300% increase in time spent in physical
activities also resulted from deep diaphragmatic training.
Related to exercise, a 2011 study looked at the impact on post exercise recovery of endurance athletes and concluded that there is a benefit.
Overall, these data demonstrate that relaxation induced by DB
increases the antioxidant defense status in athletes after exhaustive
exercise. These effects correlate with the concomitant decrease in
cortisol, which is known to negatively affect antioxidant defenses,
and the increase in melatonin, a strong antioxidant. The consequence
is a lower level of oxidative stress, which suggests that an
appropriate recovery could protect athletes from long-term adverse
effects of free radicals.
I've personally had a coach that encouraged belly breathing during physical exercise (bicycle racing). In that sport you often have periods of little jarring, fairly smooth riding, and being able to operate for long periods of time at a rather high power output (without going anaerobic) is key.
Anecdotaly, the other thing I can offer up is that when running I try to time my breaths with my steps, if I'm holding a given pace. It just seems that mechanically there are times in your foot-strike->airbone->foot-strike pattern that are better for intake and exhalation than others.
And the Valsalva method is typically employed for lifting heavy weights. You'd do it naturally: before you pick up a heavy object most people will take a breath, hold it, and lift.