I forgot to mention what is probably the most common use here. Dips.
Almost every meal is served with a dip of some sort and they nearly always contain Chinese chives. Here is a simple dip served with jiaozi dumplings. It's just soy sauce and chopped chives.
More common is this. It is served with all meals. Chinese chives, coriander leaf (cilantro) and hot red peppers are chopped finely and mixed with soy sauce and/or vinegar. So common is this that my nearest supermarket only sells the coriander tied in a bunch with chives.
Chive and Coriander Bunch as sold in most markets and supermarkets for making dips.