Divorced, two kids, shared custody - who can file what on their taxes?

In the absence of a written agreement the answer, stated coldly, is the one with possession of the body gets the write-off.

No person can be claimed on more than one return. You can agree between yourselves who will claim each child. If you can't agree, the parent the child lived with more nights of the year gets the claim, or if that's exactly equal, the parent with the higher AGI.

The parent who claims the child gets the child tax credit. The parent the child was with more nights of the year, even if that's not the parent who claims the child as a dependent, is the only one eligible for EIC or dependent care credit for the child. Both parents can't claim the SAME credit for the SAME child.

Child support payments aren't deductible for the payer or income to the recipient - they have nothing whatsoever to do with your taxes.

This situation is usually covered in the divorce decree. If it wasn't, the most common solution if for them to alternate years for claiming the dependent exemptions, eic, child tax credit, etc.

If they can't come to an agreement, the IRS has a series of tie breaking rules, which are outlined on page 28 of IRS Pub 17:

With everything else being equal - they lived with each parent the exact same amount of time, the exemptions goes to the parent with the higher adjusted gross income.

Child support is not deductible by the payor nor is it taxable income to the recipient; and it does not affect EIC. You should make sure the divorce decree specifies who gets to claim who and yadda; and that the other party is court ordered to sign whatever release is necessary in order to facilitate a tax filing. Both returns have to be correct in order to keep the IRS happy; and this happiness rolls downhill.

should be in your custody agreement, talk to a lawyer about what you guys are to do if this hasn't been brought up;; but read your paperwork, should already be in there...