While everyone lives in one home, your mother files her income on her tax return and you file your income on your tax return. it's that simple.
As for the Filing Status, there are two critical rules to filing HOH
1) you must provides more than 50% of the household expenses,
2) you must be able to claim a Qualifying Person.
So it is fine for you to file Single and claim your child
however,
short of more family members you did not explain in your post, your Mom can't file HOH unless she meets those two requirements.
if you have permitted your mother to claim your child, then
1) you are filing SINGLE with no dependents
2) and assuming your mother provides more than 50% of the household expenses, she files HOH and claims your child.
Unless there are other qualified persons in the household that you did not mention, your mother can't file HOH with no dependents.
