I've started adding various greens (spinach and arugula) to my eggs and omelets. I've only tried a few ways:   Add greens first to a hot pan with olive oil and butter. Add the eggs afterwards. Cook the eggs, etc. first then add the greens near the end of cooking.   Ideally, I'd like to preserve the freshness of the greens without it being a salad on top of my eggs.   Also, keeping the leaves whole doesn't seem to work out the best. Perhaps I should chop the greens before adding? 
Well the traditional french way would be that you add spices to eggs before cooking and stuff the omelette right before serving.   I've watched a few youtube videos on the matter and I like how Jacques Pepin explains the process.  Check it here.  Jamie Olivers version is quite similar, he stuff omelette with cheese before he folds it. Check it here.  I've tried all version and I like them all. The main trick about an omelette is that you do not overcook the eggs. Everything else is personal preference I'd say, like whether to chop or no the greens.  