I'd say it REALLY depends on the tone you're going for and the exact feeling you're attempting to reproduce in the story.
So, "sigh" and "gasp" are probably the most unobtrusive to readers… they're the ones that are the least noticeable when you're reading and the most easy to understand. They tend to be the best in average, every-day situations. However, both "sigh" and "gasp" have emotional connotations— a sighing character is usually sad or reflective, a gasping character is surprised.
For action description like "deep breath" or "exhale", you've got a little more leeway with emotional connotations, so they tend to be good for situations where "sigh" or "gasp" is too specific. A "deep breath" can mean the character is angry (and about to yell a lot), surprised, quietly reserved to events, scared. "Exhale" can likewise be paired with almost any emotion. (I wouldn't consider "shocked" to be an action description because that's an emotion, not a type of breathing)
And onomatopoeia is always fun! I'm with Anna, I tend to use "haaaaaah" as a sighing sound. I'm not sure if "phew" is onomatopoeia, but I use that one a lot, too! Onomatopoeia can be used for comedic effect (it tends to look the goofiest spelled out) and so it's often used for lighter and funnier scenes. But it can also be used for dramatic effect— especially repetitive sounds, like "huff huff huff" or… uh, I'm drawing a blank. But you get the idea! Sound affects tend to be the easiest of these to customize visually, too, with unusual fonts and colors etc, so they can also use that for emotional punch.
So basically what I'm saying here is… there's no wrong way to go about it?? I'm pretty sure I've used all of these in the same comic, haha. Just try to weigh what the emotions each word you choose is going to express (reading aloud is great for that) and decide which one works the best in the context of your scene.