- "...polystrate trees (trees extending through two or more strata, each of which, according to evolutionary interpretations, was deposited slowly over a long period of time)."
--Duane Gish, Creation Research Society Quarterly 12(1):34-46 June, 1975
- "Fossils that cross two or more sedimentary layers (strata) are called poly (many) strate (strata) fossils. Consider how quickly the tree trunk must have been buried. Had it been slowly, its top would have decayed. Obviously, the tree could not have grown up through the strata without sunlight and air. The only alternative is rapid burial."
--Dr. Walt Brown, Center for Scientific Creation
The basic claim, then, is that these fossils cannot be explained by conventional geology. The argument rests on two planks. The first plank is that trees rot quickly. The second plank is that conventional geology is uniformitarian. Therefore, scientists allow only slow, steady deposition, and cannot explain catastrophic events.
It is then argued that this rapid burial is best explained by the biblical flood. So, it is argued, Creationism has an explanation for polystrate fossils, and conventional science does not.
This argument has been made by many Creationist authors, including Coffin, Huse, Morris, Taylor, and Gish.
These fossils are reasonably common, and have been mentioned in the scientific literature for well over a century, under the name upright fossils or in situ trees. No well-read geologist finds them surprising, and no geologist has ever claimed that it took millions (or even thousands) of years to bury them. Science is perfectly happy with the idea that deposition is occasionally rapid.
Geologists agree that the numerous upright fossils couldn't have all been buried at the same time.
There is a Usenet FAQ on this.
Big trees commonly stay upright for years and even decades after death. Some "drowned" trees can even keep living, if they are tolerant of waterlogged conditions. For example, at Reelfoot Lake in Tennessee, there are some bald cypress trees that have kept growing in the lake since an earthquake submerged them in 1812.
Some upright fossils had rotted-away interiors by the time their burial was final. So, in those cases, the tree may have stood dead for some time. The typical height of upright fossils is on the order of two meters, so many of these fossils represent only the base of the original tree. The top of the tree presumably rotted.
To tell this, geologists look at the material surrounding the fossil.
It is possible the material is a sandstone made of wind-deposited sand. Some sandstones are essentially fossilized sand dunes. We know that in the modern world, sand dunes move fairly quickly. For example, the Sleeping Bear Sand Dunes, on the eastern shore of Lake Michigan, have been known to cover trees ten meters high in just a few years.
It is much commoner for these fossils to be buried by volcanic ash, or by a debris flow caused by an eruption. As Mt. St. Helens and Mt. Pinatubo showed, a single eruption is all that's needed, and even an intense pyroclastic flow does not flatten all of the trees. The Tertiary Period fossil forests in Yellowstone National Park, USA are of the volcanically-buried type.
It is even more common for these fossils to be found in sandstone or mudstone which gives evidence of being river-deposited. A rock of this type sometimes has layers which are each several meters thick. Many of these upright fossils are in Carboniferous Period rocks with coal deposits. There, they are often rooted in the top of the coal seams or in fossil soil deposits, and are buried in an overlying sedimentary rock. The upright fossils of Joggins, Nova Scotia, Canada are of this type.
In the present day, deposition just like this can be done by floods, by natural levee breaches, and by course changes of river channels. Such floods often repeat themselves several times per decade, particularly in a basin area which is sinking. Each repetition would leave one layer.
All known upright fossils were buried in days, a year or so, or else periodically across perhaps a few decades. They occur all over the world, because swamps, river deltas and volcanoes also occur all over the world.
Yes. For example, the coast of Nova Scotia has a marsh which is sinking, and burying the trees in it. Wood buried in such marshes is almost immune to rot. This is happening because there are regions all over the world which are sinking, or rising. People who study Roman ports (or Etruscan ones) often find that the ancient port is a few meters higher or lower than the current sea level.
When Mount St. Helens erupted several hundred years ago, the debris flow buried trees. Those buried trees are just like the ones found at Specimen Ridge in Yellowstone National Park. They have already started to mineralize into "petrified logs", but it was still possible to carbon date them. 
There are at least three lines of argument against this.
The first argument is that the fossils aren't all found on one single level of the Geologic Column. Some are from the Devonian Period, well before the dinosaurs. Some were buried long after the dinosaurs went extinct. This is what you would expect if each burial was caused by a small, local event. And, there are differences, depending on where they are found. For example, giant lycopod trees are only found in Carboniferous Period rocks, and cypress trees aren't found below the Cretaceous Period. The same comment applies to their leaves and spores and pollen. But this is exactly what you would not expect if a single, global flood had washed over them. Surely the flood would have ripped many trees up, and dropped them elsewhere. Or if not the trees, at least the pollen.
The second argument is that some upright fossils were transported to where they are now. Others are clearly still in place (in situ), because they are still rooted into a fossilized soil. The transported trees have had their root systems ripped, but the in situ trees still have the small, fine rootlets in place. It does not seem possible for a single global event to transport some trees and not others.
The third argument is that there are some upright trees which are on top of other upright trees. We know that the upper tree grew after the lower one was buried, because the uppper tree is clearly in situ.
An example of this is a burrow pit near Donaldsonville, LA. When they excavated backswamp clays to rebuild the adjacent levee, they uncovered three levels of upright cypress forests buried on top of each other beneath the recent floodplain. These polystrate trees are buried within recent Mississippi River deposits that are only 4,000 years old. The much older upright trees in Yellowstone Park are similarly layered.
The following references are selected from the ones posted to Usenet's "talk.origins" by Andrew MacRae and Keith Littleton. More can be found in the Usenet FAQ.
 Early Pennsylvanian swamp forests in the Mary Lee coal zone, Warrior Basin, Alabama, Gastaldo, R. A. (1990) in Carboniferous Coastal Environments and Paleocommunities of the Mary Lee Coal Zone, Marion and Walker Counties, Alabama. R. A. Gastaldo and others, Guidebook for the Field Trip VI, Alabama Geological Survey, Tuscaloosa, Alabama. pp. 41-54.
 Sonar and scuba survey of a submerged allochthonous "forest" in Spirit Lake, Washington, Coffin, H.G., 1987. Palaios v.2, p.178-181.
 Erect floating stumps in Spirit Lake, Washington, Coffin, H.G., 1983. Geology v.11, p.298-299.
 Tree-ring dating of pre-1980 volcanic flowage deposits at Mount St. Helens, Washington, Yamaguchi, D. K., and Hoblitt (1995). Geological Society of America Bulletin, vol. 107, no. 9, pp. 1077-1093.
 The recent upright trees of Mt. Hood, Lawrence, Donald B., and Elizabeth G. Lawrence (1959) Mazama vol. 40, no. 14, pp. 10-18.
 Prehistoric buried forests of Mount Hood. Oregon, Cameron, Kenneth A., and Patrick T. Pringle (1991), Geology vol. 53, no. 2, pp. 34-43.
 Burial of trees by eruptions of Mount St. Helens, Washington: Implications for the interpretation of fossil forests, Karowe, A.L. and Jefferson, T.H., 1987. Geological Magazine v.124, no.3, p.191-204.
 The fossil cliffs of Joggins, Ferguson, L, 1988. Nova Scotia Museum, Halifax, Nova Scotia.
 A classic Carboniferous section; Joggins, Nova Scotia, Gibling, Martin R. 1987. In Centennial Field Guide, Volume 5, Northeastern Section of the Geological Society of America Roy, D.C. (ed.), p. 409-414
References for YellowStone National Park are in this Usenet FAQ