How to Know God Exists, in which Ray discusses the fossil record, is that Ray is given to quoting experts out of context, with quotes about fossils in general substituting for actually discussing specific fossils or their features The most important fact, though, is probably that he does not have (or apparently want to have) any clear idea of what "species" means to a biologist or paleontologist. He also doesn't seem to have any clear idea of what a "transitional form" ought to be, or what we ought to expect from the fossil record if evolution is true.
Ray, for example, cites a figure that there are 250,000 known fossil species. Ray does not ask how many living species are known (at least seven times that number), or how many of those fossil species are identical to living species and how many are extinct, or how many fossil species are known from a single incomplete specimen. If he did these things, he would be led to the conclusion that most living (and by extension presumably most extinct) species have no fossil record, which in turn implies that we ought to expect, if evolution is true, that the transitional fossil record will be full of gaps.
He notes that living things fall into distinct (allowing for some fuzziness around some species) species, rather than an unclassifiable continuum of living things, a point raised by Darwin in On the Origin of Species, though Ray doesn't mention Darwin's suggestion that this is because the intermediate forms died out, leaving only distinct, well-adapted groups. And as noted, he tries not to discuss extinct intermediate forms.
Ray mentions only a few transitional fossils. The hoax "Archaeoraptor" is mentioned as the only feathered dinosaur; Ray notes neither the existence of, e.g. Sinosauropteryx or Caudipteryx or Anchiornis (that last, to be sure, may have been discovered too late to feature in the book), nor does he note that "Archaeoraptor" is a composite of two real (and genuinely transitional) fossils, one of Microraptor (a genuine feathered dinosaur, almost but not quite so birdlike as Archaeopteryx) and one of Yanornis (a primitive bird with a toothed bill). He likewise doesn't note that Archaeopteryx itself is, skeletally, so like non-bird coelurosaurs that one of its fossils was classified as the theropod Compsognathus until faint flight feather impressions were noted on the fossil. Perhaps Ray is waiting for a dinosaur that has one modern bird wing on one side, and a scaled, clawed arm on the other, before he will acknowledge a "dino-bird transitional."
Ray mentions the famous horse series only in the context of a quote noting that its individual species appear and vanish without transitional forms linking them to other species. This is of course an argument for punctuated equilibrium -- speciation over a few hundred or thousand generations in small, local populations. It isn't an argument against speciation as such because we know (and most creationists accept) that speciation is possible. Scientists have observed, e.g. the fruit fly Drosophilia melanogaster give rise to descendants of D. paulistorum. Polar bears can not only interbreed with brown bears, but genetically are more similar to some brown bear populations than to other brown bear populations, convincing even most creationists that they did indeed evolve from brown bears. "But they're still fruit flies and bears!" will of course be Ray's retort, but then, by the same token, all those distinct species of Mesohippus and other equid genera are just transitional forms between Eocene condylarths and modern horses. We know that transitions between species are possible, and the species themselves form transitional series between genera and families within the hoofed mammals.
The discussion of the hominin fossil record is of a piece with the above. Ray speaks of Homo erectus (well, actually of "Java man") as though the Trinil skull cap and thigh bone were the entirety of the H. erectus fossil record. He speaks of Neanderthals as though the Old Man of La Chapelle-Aux-Saints were the only specimen, rather than one of a couple of hundred, including children and infants. He gives space, of course, to Nebraska Man and Piltdown, but not to the Dmanisi skulls or Turkana Boy or ER1470. There are no "ape men" if one insists that "ape men" like H. erectus are fully human and if one simply ignores more primitive hominines and australopithecines.
There's a brief mention of the Cambrian "explosion," with quotes from assorted scientists. There is, of course, no mention of Precambrian fossils, or Ediacaran (late Precambrian) representatives of modern phyla like jellyfish and sponges, or the "small shelly fauna" that make up the first ten million years or so of the Cambrian fossil record -- bits and pieces of hard exoskeletons before full exoskeletons appear, as if by magic (or,alternatively, as if their body coverings evolved from hard bits and pieces to full shells and carapaces, showing off their full shapes better). And from this cursory and extravagantly superficial discussion of the fossil record, Ray concludes that it contradicts the predictions of evolutionary theory (although, considering that he seems a bit vague on the whole concept of common descent, it's not clear that he has any real concept of what the theory predicts).
The fossil record supports common descent with modification. Quote-mines about the fossil record support Ray Comfort, though not very strongly.