Which fresh herbs goes well with an omelette?
Plain omelette: parsely and chives.

I'd be careful with woody, stalky herbs like rosemary; not because of the taste but because I don't want to chew on the hard pieces. Keep in mind that the omelette does not get cooked as long as other dishes where these work well.

Then depending on the filling: anything with cream and fish may work with dill. Italian style with sage and thyme. (Not herbs, but for completeness: Paprika with mushrooms and red wine. Garlic with fried onions.)