The LDS have been digitizing the microfilm collections and other records (with permission of the record owners) and placing them on their website, FamilySearch. Find the digitized collections from the homepage by clicking on search, scrolling down to browse by location and then clicking on the region you desire. So after you select United States, either scroll down the list of databases to find Michigan or in the left column scroll down to click Michigan to see just the Michigan databases.
Here you will find:
Michigan, Births, 1867-1902 (index and record images, from Michigan Department of Vital and Health Records)
Michigan, Marriages, 1867-1925 (index and record images, from Michigan Department of Vital and Health Records)
Michigan, Deaths, 1867-1897 (index and record images, from Michigan Department of Vital and Health Records)
Michigan, Death Certificates, 1921-1952 (index only, from Michigan Department of Vital and Health Records)
Michigan, County Marriages, 1820-1935 (index and images)
Michigan Death Index, 1971-1996 (index only from Michigan Department of Vital and Health Records, provided by ancestry.com)
There are also a couple of index only databases (Births & Christenings, Marriages, and Deaths & Burials) that are comprised of former parts of the International Genealogical Index (IGI) or Vital Records Index collections. You would need to check each entry to determine the original source of the information (extract from an original record or submission from a person).
SeekingMichigan - a FREE website of the Archives of Michigan
The Archives of Michigan's SeekingMichigan website is where you will find the Michigan Death Certificates Collection 1897-1920. These records are from the Michigan Department of Vital and Health Records. Use the Advanced Search and only search one database at a time in order to get filters specialized for the death certificates. There is no Soundex option in this search engine.
According to our speaker from the Archives of Michigan back in November and the latest Michigan Genealogical Council newsletter, hopefully within the next three to six months the Michigan Death Certificates 1921 through 1938 will be added to the SeekingMichigan site. Additional years will be added as the law allows.
Ancestry.com/Ancestry Library Edition - a $ PAY-service via home or library subscription
Ancestry is a commercial website. If you don't have a home subscription, then come into the Mt. Clemens Public Library and you can use the library's subscription. (Or check to see if your local library has a subscription.)
On Ancestry's website, you'll find the following index databases under Michigan:
Michigan Death Records 1897-1920 is an index to the death certificates found at SeekingMichigan. This database index has the Soundex search option and links to take you to the images on SeekingMichigan.
Michigan, Deaths, 1971-1996 (index only, original data from the Michigan Department of Vital and Health Records)
There are a couple other index only databases for vitals records that are based on the index only databases located at FamilySearch. Again, check where the information came from for each entry.
Related to civil vital records are church vital records. Ancestry has a database with index and record images called Early US French Catholic Church Records (Drouin Collection, 1695-1954). It contains records for a variety of states. For Michigan, there are several Catholic churches, including records earlier than when the State of Michigan began recording vital records.