I think when the Supreme Court has a 5-4 decision on an important law, like capital punishment or abortion rights, then the dissenting opinion becomes vital. A future case may very well threaten to overturn the original decision, so it's important to know where certain justices stood on the issue. Knowing if the decision fell along liberal and conservative lines might help predict future behavior.
The dissenting opinions I find interesting are the ones issued when the vote is more like 7-2 or 8-1. The dissenting judges are almost always identical in political persuasion. If a law protecting workers' rights is upheld, I can almost guess the dissenting opinion will be written by a conservative justice. If a decision involving Second Amendment rights is upheld, I can bet the dissenting opinion will be written by a moderate to liberal justice.