In the U.S. and Canada, it’s customize, customized, customizing, customization, customizable, etc. Outside North America, it’s customise, customised, customising, customisation, customisable, etc.
Customize is a mid-20th-century American creation, so the z spelling is the original. The word is so new that some dictionaries, including the main British ones, have yet to recognize customise etc., but that spelling is now very common in all sorts of texts from outside North America. In recent British news stories, for instance, customise appears about 20 times for every customize. In Canada and the U.S., meanwhile, customize appears about 500 times for every customise.