The key to keeping custody of your child is: document, document, document. That is, get court, police, and other records for as much of the father's misspent life as you can. You'll need these records whether you go to divorce court (you don't mention whether you married him or not) or a paternity court (which handles custody and support for unmarried parents).

Can you just waltz in and get a copy of someone's police record? Not exactly. Police and courts have different requirements for giving someone in your position records of arrests and crimes. Try going to the places where you suspect this guy has been arrested or brought to court. Look for the records office, hang out in the back of the lobby, and watch the clerks until you figure out which one is the most sympathetic. Then explain your predicament and ask what the clerk can give you to take to court. They might, um, bend a few rules for you.

For criminal incidents where you can't find an official record, see whether you can find a victim of one of the crimes to give you a handwritten note describing what happened. It doesn't need to be in fancy legal language, but should give as much detail as possible. Just above the victim's signature, he or she should write "I declare under penalty of perjury the foregoing is true and correct." Don't forget to add the date, too.

As far as the father's visitation and (lack of) support, get a little pocket calendar and write down exactly when he visits and how much money he gives you. Give him a receipt for every payment to show that you are a responsible bookkeeper.

After you've collected these various records, make several copies to leave with the agencies you may have to deal with. (But try to hang on to an original and/or certified copy for future use.) Once you have these documents in your hands, you'll feel much stronger when you get to mediation or court. After all, it won't just be your word against the father's.