Hello!
I care that the designs (XHTML/CSS) are valid (validator.w3.org). Then most browsers- even IE7- are working quite well. Using a text browser like Lynx is often also very interesting.
Well, there are many, many browsers out there- but being tested with a current Opera, Firefox and Internet Explorer version should do it I think.
Very useful is the tool "TopStyle3", which is only for CSS/(X)HTML programming. It is also checking the code for nearly all browsers available and informing very well about possible problems. It even has a CSS preview- but of course it is too expensive for the casual web designer.
Yours Michael