In its most general sense, Creationism was a common early account of how the universe came to be. It was based on an analogy of the sort "if you see a tent, you imagine a tent maker". In other words, the existence of a thing, especially a complex thing, presumes a creator with a purpose. This general form of creationism, still promulgated by advocates of intelligent design, including distinguished scientists like John Polkinghorne, is not incompatible with modern science or with evolutionary theories. It assumes a divine will behind our universe as scientists perceive it, causing it to have come into existence and to function in a certain way. The main argument is that such a complex system shows evidence of design beyond what can be derived from our knowledge of physical processes.
Another type of Creationism argues from Biblical inerrancy, claiming that (1) there is good evidence for the truth of the Bible; (2) if the Bible is true and must be interpreted literally the we must accept Genesis as a literal account of the origin of the earth and its creatures. It should be noted that (1) does not entail (2).
What sort of arguments have been offered to support theories of Creationism and Intelligent Design?
In a nutshell... bad ones. None of the arguments offered for ID withstand scientific investigation. It is simply a false proposition that cannot be supported.