Stabilizing selection "acts" against change by favoring intermediate phenotypes.¹ Many suspect that this is the stasis that Niles Eldgridge and Stephen Jay Gould noticed in the fossil record and serves as a central element in their theory of Punctuated Equilibrium. It is important to note that selection is still operative, it simply reduces the fitness of those at the tails of the distribution, canalizing the species toward a particular phenotypic configuration (think more metastable as opposed to static).

As I implied above, some paleontologists have observed that there are sharp discontinuities in the fossil record between species which obviously have a phylogenetic relationship. One common inference is that the rate of evolution (sharp fitness differentials between phenotypes tracked by differences between genotypes resulting in greater selection response between generations) sped up to the point where intermediate species were not preserved (perhaps due to catastrophic environmental changes). But there is another more prosaic explanation: allopatric speciation, where a contiguous population becomes separated by some barrier which allows the accumulation of genetic changes and eventually fosters speciation. Imagine that a continent bifurcates into a large fragment, "A," and a small fragment, "B." Now, consider that two populations of a species are separated, and population A and B eventually fix alleles at various loci which results in speciation. Consider the following scenario: continent A and island B are reconnected by an isthmus. Species A now drives species B, which has been isolated and not subject to as many varied selection pressures, to extinction in short order. Eventually, species A also becomes extinct. Over the eons most of the reunited continent is subject to geological change, except for the portion that was once island B. When paleontologists analyze the remains in the region that was once island B, they will find that an ancestral species that evolved into the form of population B. But at some point there is a "jump" where population A appears in the fossil record and B vanishes almost simultaneously. Since A is likely to be morphologically similar, but different, from B, they might infer that the ancestral population evolved into B, and jerked into population A.²

As we've been learning, beware of some tales fossils tell.

1 - The example in the link provided was infant mortality as a function of weight, but one could easily imagine some sort of sexual selection also working in this fashion. We know of the studies which suggest composite images are often more attractive than the normal person off the street (because they are closer to the populational median?), and some have suggested that social selection preserves phenotypes even when ancestry is confounded.

2 - Yes, proponents of Punctuated Equilibrium are aware of this process and offer it as an important factor in what they are observing, but they also suggest a widespread prevelance of macromutations in nature (possibly because macromutations are more challenging to the Neo-Darwinian Synthesis).