What is better for your credit score, a zero balance or carrying a small balance?

The best thing to do is to keep balances at or near zero. While it is good to have credit card activity (ie. don't open a card and leave it unused just for the sake of having a zero balance card on your reports), maintaining a good credit score is easier if you keep balances as low as possible.

And if you want to eek out a few extra points for example if you are applying for a home loan, note when your credit card company reports to the credit bureaus. Even if you completely pay your cards off each month, your credit reports can show a balance because the credit card company can report charges that have been added since the last bill was due. Try making a payment outside of the typical billing cycle in order to get the balance reported to the credit bureaus as close to zero as possible.