In general, the plain edge is better than the serrated when the application involves push cuts. Also, the plain edge is superior when extreme control, accuracy, and clean cuts are necessary, regardless of whether or not the job is push cuts or slices.
In general, the serrated edge will work better than the plain edge for slicing cuts, especially through hard or tough surfaces, where the serrations tend to grab and cut the surface easily. Some of the cutting power of the serrated edge is due to its format alone; thus, even a dull serrated edge knife will often perform competently at slicing jobs. The serrated edge gets its slicing ability from a number of factors. The high points on the serrations will touch the material first, and this gives those points higher pressure per area than if the same pressure was applied to a plain blade; this allows the serration to puncture more easily. In addition, serrations are normally chisel-ground into the blade, which means they are thinner (and thus cut better) than the comparable plain blade.
The plain edge will work better for applications like shaving, skinning an apple, skinning a deer. All those applications involve either mostly push cuts, or the need for extreme control. Serrations work really well on things like tough rope or wood, where the serrations bite through quickly.
Generally, the more push cuts are used, the more necessary it is for the plain edge to have a "razor polished" edge. A knife edge becomes more polished when you move to higher and higher grit stones. Generally, 1200-grit is considered polished; a 6000 grit Japanese water stone would polish the edge further.
One interesting case is cutting a tomato. In theory, you can just push a blade through a tomato, so a razor polished plain edge would work fine. However, the tomato is soft, and unless your plain edge knife is very sharp, the tomato will simply squish when you start pushing. You can (and many people do) use a slicing motion with your plain blade, but if it's even a little dull it won't cut well and it may not even break the skin. Use a sawing motion with a serrated knife (even a dull one), and your tomato will slice fine.
You will read about test after test where the above view is confirmed. That is, the plain edge excels in push cuts, and the serrated excels in slicing cuts. This confirms the conventional view... to an extent.