Don't forget AMD's K6 processors -- these are also i586 processors.
I have an AMD K6 that won't boot Fedora 7 (or later) due to missing some bit of architecture (I forget specifics, sorry...). So I suspect it's not truly an i586 processor? (fwiw, It did boot and install Linux Mint 9, LXDE however.)
I didn't know the difference 10 years ago when I bought it, though it had Win98 installed which was fine back then... Now it's barely adequate as a print server.
My last attempt to run a later kernel on a 586 or older was with EL 3. But if the system was rebuilt, it lost the updated native threads introduced in 3 (RH 9). The GLIBC apparently required a 686 or later to support it.