I've heard both holding your breath for all reps and exhaling on the way up - what is the best approach and/or differences in approach? and are there other breathing patterns?
For any breathing under load, the best approach is to use the same technique singers, cheerleaders, and martial artists use: breath from the gut. Look at yourself in a mirror, and if your shoulders are rising and falling when you breath you are breathing from your chest. Your stomach should be moving in and out, almost as if your diaphragm is being moved by your lower abdominal muscles.
According to "The Vault" by Dave Tate, "Starting Strength" by Mark Rippetoe and Dr. Lon Kilgore, and a few other sources:
That held breath provides a stable platform, and as soon as you move your ribcage (breathing through your chest), you will lose precious tightness. Losing tightness will affect your ability to finish the reps in your set. Your breath should only come from your abs moving, and just enough to keep you going.
I don't think it would be possible to hold your breath through a set, so I'll assume you mean whether or not to hold your breath throughout one complete rep.
Keeping your lungs full of air helps create intra-abdominal pressure, which stabilizes your core and adds support for a weight above you. I find this to be most crucial when doing squats, but certainly that extra stabilization can help with bench press. Any reduction in this pressure means a reduction in tension. Take the excerpt below from a Pavel Tsatsouline interview, in the context of intra-abdominal pressure for strength training.
That said, you have to balance with comfort, as you don't need distractions to your focus. I find myself exhaling a little during the upwards part of heavy lifts. I never exhale all the way, because I want to maintain that pressure for safety (and for the next rep, if there is one). Certainly, you only want to inhale (quickly) between reps.
Note that if you have heart or blood pressure problems then breath-holding may carry some additional risk.