We can observe in a laboratory setting any individual aspect of evolution from the genome itself all the way up to speciation. So, if we can see every aspect of evolution in a lab, what's the argument about? Two things as near as I can tell.
First, timeline. Yes we can observe all the facets of evolution, but if we don't have the requisite millions of generations to work with, it's not an explanation for all life as we know it. For this reason I believe a large number of people who disagree with evolution are actually disagreeing with the geological data we have about our earth and the astronomical data about our universe.
Second, God. If the facts are accepted as science has explained them, some definitions and views of God are tossed out the window, meaning that you cannot both accept science as it has been explored and God. Of course, this is the basis of faith - accepting the reality and truth of god regardless of fact - accepting God because you want to.
If these premises were true we would expect the majority of people who reject the claims of evolution to fall into one of these two categories, and while there is a lot of debate about details and finer points of evolution, the only people who outright reject the general patterns of evolution as an answer to the diversity of life tend to fall into one of these two camps.
In reading another article from Answers in Genesis I found that Ken Ham agrees with me. Specifically, "That is, both of the above groups suffer from the same basic problem. They really don’t understand that it is not a matter of ‘their evidence vs ours.’ All evidence is actually interpreted, and all scientists actually have the same observations—the same data—available to them in principle." ... "I then give an example. ‘Let’s consider the science of genetics and natural selection. Evolutionists believe in natural selection—that is real science, as you observe it happening. Well, creationists also believe in natural selection. Evolutionists accept the science of genetics—well, so do creationists.
‘However, here is the difference: Evolutionists believe that, over millions of years, one kind of animal has changed into a totally different kind. However, creationists, based on the Bible’s account of origins, believe that God created separate kinds of animals and plants to reproduce their own kind—therefore one kind will not turn into a totally different kind.
‘Now this can be tested in the present. The scientific observations support the creationist interpretation that the changes we see are not creating new information. The changes are all within the originally created pool of information of that kind; sorting, shuffling or degrading it. The creationist account of history, based on the Bible, provides the correct basis to interpret the evidence of the present—and real science confirms the interpretation.’"
So Mr. Ham does accept the same things to a point. For some reason he rejects speciation. If this is because, as he says, it is not real science, then we shouldn't have any observed instances of speciation in laboratory settings. We shouldn't be able to predict and point out where portions of speciation occurred. We shouldn't have any kind of transitional records. However, we have all of these things. So what this means is that Mr. Ham is all in favor of openly admitting to all the things evolutionary science says which do not contradict his beliefs in the Bible and its God.
Unfortunately, I can't specifically refute Mr. Ham's claims that one kind of organism went through sufficient changes to become a completely different kind as I'm unsure of his particular definition of what a species or kind is. (There are various definitions ranging from specific genetic incompatibility to behavioral incompatibility, etc...) However, if we can observe speciation at all, then the only barrier for one slowly changing into something else is time.
There are at least some repeatable speciation events, particularly everyone's favorite genetic toy, Drosphilia.
And of course there are numerous articles describing various speciation events.
However, Mr. Ham rejects this information for some reason, as well as various articles regarding transitional fossils.
Is it a simple matter of interpreting the facts differently? Or is it a matter of rejecting the facts because they don't fit into his world view? Given the fact that most of what he rejects provides a more complete and useful explanation of the world around us, I cannot help but think it's got less to do with interpretation of facts and more to do with preservation of a belief system.