This one: "it just means Dell has not tested that machine for Windows 8 and would not help with problems that may arise" Dell only supports operating systems that shipped on a particular machine.  Since Dell never shipped a 580 with Windows 8, there was no reason to test it, so they didn't.  They won't test and validate systems on newer and newer operating systems as they come out.  That said, there should be no other good reason why it couldn't run Windows 8.  The compatibility test should be your best indicator.
I wish to upgrade from Windows 7 to Windows 8. The Dell site says my machine is not supported by Dell for Windows 8. Does that mean it is not possible to use Windows 8 on the Inspiron 580, or does it just mean Dell have not tested that machine for Windows 8 and would not help with problems that may arise? The Microsoft Windows 8 compatibility test throws up very few problems for Windows 8 on my configuration. Would anyone advise me to go ahead or just accept there may be a fundamental reason for not going to Windows 8?  
Inspiron 580 Desktop purchased January 2011