I tend to prefer variable width in my own work, but there are plenty of fixed-width sites whose designs I respect (apple.com, for one). Setting aside mobile interfaces, you can safely go 1000 pixels wide. Of course, your lines of text shouldn't be that long -- you'll find few people advocating more than 100 characters on a line, especially for long blocks of text.

The short answer, then, is that the type of site you're doing is more relevant than some global preference.