Emergency Fund vs Paying Off Debt

I have thinking about this for quite awhile now. What is the better way to go?

I can "justify" both ways. An emergency fund is a great thing to have. If something goes wrong with your home or car, you have this as a back-up. Or, as is our situation, one partner's hours are cut at work, you have a back up until things settle down.

On the other hand, paying off debt is huge. It's not paying 19% monthly interest; it's not having that weight hanging over your head. It's having the freedom to use the credit card in case of emergencies and then pay it off.

What are your thoughts on this one?