Q. I live in Michigan and work in Ohio. How do I file the Ohio city tax?
A. Maybe not at all. See last paragraph.
Ohio has a reciprocal agreement, on wages, with all neighboring states. No Ohio state taxes are withheld or due and you do not need to file an OH return.
However the reciprocity agreement does not apply to local city taxes. You are not required to file a local city return, but you are also not entitled to a refund of the city withholding (box19 on your W-2), unless there was an error in the amount.
Michigan will give you credit, on the MI state return for OH city tax withheld, but you have to attach a copy of the city return. So, even though the Ohio city does not require you to file a return, you have to file one to get the MI credit, even though there will be no tax due or refunded from the Ohio city.
How you file an Ohio city return, depends on which OH city it is. Some use a state wide agency (see other reply). Some do it themselves. A couple of Mich residents have reported, in this forum, that they got the credit without attaching an OH city return.
