The custodial parent may claim everything child related UNLESS they waive the dependency exemption to the non custodial parent via a form 8332.... in that case the child may be used on 2 separate returns but only in the following way :
Only the Custodial parent can claim: (Child would be listed as non-dependent EIC & CC only)
-Head of Household
-Earned Income Credit
-Child Care Credit
The non custodial parent can only claim: (Child would be listed as dependent)
-The Exemption
- The Child Tax Credit
See Special rule to divorced or separated parents (or parents who live apart) on page 32:
https://www.irs.gov/pub/irs-pdf/p17.pdf
