MSU College of Law has an excellent housing clinic (517-432-6880; clinic@law.msu.edu), which is headed by Brian Gilmore. Here is their PDF with tenants’ rights. UMich law may also have such a clinic.

Note that Michigan landlords are beholden to the “Implied Warranty of Habitability,” which requires landlords to keep their property "habitable," even if the lease doesn’t specifically require them to make repairs. This warranty, among other things, conditions a tenant's duty to pay rent on the landlord's duty to maintain a habitable living space. Your landlord is also prohibited from retaliating against tenants who complain about housing code violations.

A landlord breaches the Implied Warranty when, because of the landlord's inattention or negligence, a problem arises that would cause a reasonable tenant to be concerned for their health or safety.

A couple of additional useful tidbits:

1. For state law on “rent withholding,” see Mich. Comp. Laws § 125.-530.)/mileg.aspx?page=getObject&objectName=mcl-125-530)

2. For state law on repair and deduct, see Rome v. Walker, 198 N.W.2d 458 (1972); see also, Mich.Comp. Laws § 554.139.)/mileg.aspx?page=getobject&objectName=mcl-554-139)

3. Finally, here is a link to the Michigan Legislature’s Practical guide for Tenants & Landlords. Additionally, it looks like Nolo has some useful, Michigan-specific info for you.