If a parent sells a child's ss #. She didn't work all year. But sells it to her mother(grandmother of child) to use on her income taxes. what can happen?

The penalty for filing an intentionally false tax return is to pay the taxes back, with a monetary penalty.  If the fraudulent tax return claimed earned income credit, the taxpayer can be banned from claiming EIC for the next two years.

However, there are a lot of facts missing and the situation may not be illegal.

If the child, mother and grandmother all live together for at least half the year, then the grandmother can legally claim the child as a dependent if the grandmother earns more money than the mother.

If the child and mother live separately from the grandmother, and if the mother does not file a tax return (did not earn enough to be required to file one and does not file one to claim credits or benefits), then no one can claim the child as a qualifying child dependent.  However, there is a second kind of dependent called a qualifying relative.  A child can be a qualifying relative dependent of an aunt, uncle, grandparent or non-custodial parent but only if that person pays more than half of the child's living expenses and only if the child is not a qualifying child dependent of anyone else.  Since only one person can pay more than half of the child's support, only one relative could qualify to claim the child as a qualifying relative dependent.  If the child's support is paid by several people or if some of the child's support is paid by the government (welfare, section 8, etc) then maybe no one pays more than half.  In this case it is possible to assign the dependent to one person but only if everyone who paid partial support signs an agreement.

However, a qualifying relative dependent does not allow the taxpayer to claim earned income credit or use head of household filing status.  The taxpayer can only claim the dependent exemption and the child tax credit.