I think these errors happen with different browsers for different reasons, which is what makes it so hard to track down.
IE8 Bug
Microsoft have said a bug in IE8 will (in some circumstances) generate spurious requests to the server, that do not affect the user but do lead to errors being logged at the server side.
See this discussion here: Bug IE8 – 4K dropped - "Invalid viewstate" when loading ScriptResource.axd (edit: link now requires login for some reason, sorry)
... particularly EricLaw-MSFT's update when he says:
It is worth mentioning that anyone who is experiencing a problem here in IE6/IE7 or Firefox is encountering a different problem that is not related to the IE8 issue described below.
This blog post also describes the bugs: Bugs in IE8's Lookahead Downloader
They say changing the way you set Content-Type will help with some of the errors, although not all of them - they say it is caused by various obscure circumstances that they are still looking at.
Update: As of 01/Apr/2010, these IE8 bugs have been fixed, via IE8 Cumulative Update (KB980182).
This post: IE8 Lookahead Downloader Fixed gives more detail on the bugs and other possible/partial workarounds (e.g this one) other than waiting for everyone in the world to download the fix.
Other Browsers
Haven't figured it out yet, but other browsers are also generating these errors, presumably for different reasons.
Web Farms
This problem is not restricted to sites running on web farms, but if you are running a farm, check out this answer by jesal which may help