In my experience the only reliable way to test compatibility is by doing it by hand.

Addons like IE tab for firefox/chrome that simulate other browsers work decently well, but have some shortcomings. https://addons.mozilla.org/en-us/firefox/addon/ie-tab-2-ff-3...

Compatibility testing often comes down to a compromise of picking the fewest combinations that cover the highest percentage of your audience. We tend to focus most of our time testing chrome for webkit (covers 99% of safari and opera issues), firefox, and IE on mac and pc. This covers the vast majority of our users, but we have to accept the fact that certain issues will slip through on less widely adopted browsers. Also, dropping support for IE6 cut our compatibility testing/fixing time dramatically.