Who can claim the exemption and credits depends on who is the custodial parent. (By the IRS definition of custodial parent for tax purposes - this is not the same as the custody that a court might grant.).
The test that the IRS uses to determine the custodial parent is where the child lived for more than 1/2 (or greater part) of the year. The IRS will go so far as to require counting the nights spend in each household - that person is the custodial parent for tax purposes (if exactly equal and more than 183 days - The custodial parent is the parent with the highest AGI, if less than 183 days then neither parent has custody). That can usually only occur if both parents lived with the child at the same time. And yes they are that picky.
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
But only if specifically specified in a pre-2009 divorce decree, separation agreement or the custodial spouse releases the exemption with a signed 8332 form - after 2009 the IRS only accepts a signed 8332 form that must be attached to the non-custodial parents tax return.
