Q. Do I have to claim my 18 yr old daughter (student and supported by me) on my tax return?
A. No, but there is seldom a reason for you not to. Even if you don't, she cannot "claim herself".
Q. Can she file on her own to receive the education credit?
A. Simple answer: No.
While technically there is a provision that allows your student-dependent to claim a federal tuition credit, from a practical matter it seldom works out. A student, under age 24, is only eligible for the refundable portion of the American Opportunity Credit (AOTC) if he/she supports himself by working. She cannot be supporting herself on student loans & grants and 529 plans and parental support (as you indicated she is). It is usually best if the parent claims that credit.
If the student actually has a tax liability, there is a provision to allow him to claim a non-refundable tuition credit. But then the parent must forgo claiming the student as a dependent, and the $500 other dependent credit. The student must still indicate that he can be claimed as a dependent, on his return. This is worth up to $2500 (AOTC shifts to all non refundable)
