If your ex leaves without notification or refuses to provide a forwarding address, there are a number of steps to take to find her.
Start with the easiest, and work your way down. You can do these simultaneously if time is of the essence:
First and foremost, remember that Google is your friend. A comprehensive Google search can often turn up all sorts of leads (as well as other interesting things).
If the move was within the last couple of days, check the trash at the last known residence. Once it is on the street, it is generally deemed to be in the public domain. Apartment complex trash bins are fair game as long as the management or residents don't complain (so pick a time when the office is closed and there are not likely to be many complaints). It doesn't hurt to wear old clothes and rub a little dirt on yourself. If you do that, fill a bag up with aluminum cans as you search.
Send your ex a letter at the last known address. Mark it clearly "ADDRESS SERVICE REQUESTED" (write this in bold letters and then underline it with a red pencil). If she has left a forwarding address at the post office, it will (hopefully) come back marked with the new address on it. Make sure you include a real letter, however. The post office has been known to make a mistake and accidentally forward the mail, or your ex may still be checking the old mail box.
If the child is in school, check with the last known school and see if they have a record of the forwarding school. Make sure you check your state laws on access to school records, as well as the federal statutes. Many states have very specific statutes granting non-custodial parents full access to school records.
If you know what city your ex is in, you can send letters to every school in the area requesting records. Make each letter sound like it is the only one so they don't know you are just fishing. It is generally a good idea, but not absolutely necessary, to include a copy of the birth certificate or court papers showing that you are the parent, along with a copy of the state and federal statutes that provide you with full access to the records. It can significantly reduce the delays.
Another thing you can do if you know what city your ex is living in is to check with the utility companies in the area you think they moved to and ask about the status of your water bill because you "haven't received one for a while". Ask them to "verify" the address they have on file- usually they'll read you the address you're looking for right off their computer screen.
Try to think of any other records you can access that would show the new location. Think of everything you put your name and address on. Be creative- there are lots of ways to gather information. Some are more ethical than others. Go through your own bills to see if they give you any ideas. Realize that you know all of the keys to getting this information, such as mothers' maiden name, social security number, birth dates, and more. Use what you know to find what you don't.
Try some of these free or low cost resources on a regular basis. You are unlikely to find anything at first, but as time passes and databases get updated, your chances of success increase: