Filing fees must be paid or waived by the court in all cases. Fees, and other costs, must be paid by the client. The rules of ethics for lawyerss prohibit attorneys from paying fees for clients; however, fees may be advanced by the attorney, but only by prearranged agreement.
A court has the authority to waive the filing fee in a procedure called an application for in forma pauperis. This fee waiver is only granted in cases of financial hardship, and clients should not expect the judge in their case to grant this fee waiver.
Fee schedules vary from county to county.
Here are the schedules of the filing fees for Ramsey County Family Court and Hennepin County Family Court, plus a link to the web site for the Minnesota District Courts, where all of the fee schedules are located.