Some of the minutiae will vary from state to state. But in general the plain view doctrine holds in all states.
In some areas, you could look at serial numbers on stereo gear if you knew that the suspect had been arrested for stolen property and that that particular gear had been reported stolen in the area. Say the local Circuit City had been burglarized (you had been the investigating officer maybe) and several Whizbang 7000Xs had been stolen and here in plain sight are a few Whizbang 7000x boxes and one set up. The boxes would have serial numbers probably visible and if one of those hit then looking at the rest. In many areas once you've found additional stolen property you could then expand the search.
On the aforementioned chainsaw, as long as you could articulate a good reason for checking the serial number it would be good, if the warrant talked about bikes only, why would you be checking the chainsaw? (ok maybe the blood stains, but other than that?)