How can I find out who claimed my child on their tax return?

You can't - tax returns are confidential.

But.. you don't need to know who did. If you feel your child was improperly claimed by someone else, you should file your tax return and claim your child anyway. You will have to file a paper return rather than e-file. You may want to include an explanation as to why you believe you are the rightful claimant of the child - for example, if you have custody, or if it's spelled out in a divorce decree you can attach that. The IRS will notice that 2 people have claimed the same child, and they will resolve the issue. See Table 6 on page 11 of Pub 501 to confirm or yourself that you are the correct person to claim the child:

http://www.irs.gov/pub/irs-pdf/p501.pdf

How can i find out who claimed my child on their tax return?
How do you know that someone claimed your child? If you are the one you can claim the child, then do it on your return.