IE8 has a IE7 compatability mode, which can be turned on an off by clicking an icon, and it remembers the setting for that site.
It's not so much a case of "Does IE8 work with xxxx site ?" - its more a case of "Is xxxxx site well enough written to work with the latest browser ?" . Microsoft warned site owners last year that a new browser was coming, and that site owners should check their site coding to ensure that it worked properly. Some site owners didn't heed the warning.
I have IE8, and there is only one site that I visit on a regular basis that has a problem that isn't fixed by the compatability button, and that is Ancestry. This problem isn't major, it just means that entering or editing the place of birth for a person causes a problem when entering the first few characters, but it works if you persist. This is the one and only problem that I have with IE8, and given the great features that it has that are not on IE7, I'm content to live with it, and hope that Ancestry gets its act together soon, and rectifies the problems with its site coding.