James G. Genco in his book Arming Michigan’s Regiments 1862 – 1864 lists what weapons were issued by dates to all Michigan regiments. He does this by number of weapons issued to each company in the regiment. He shows at on Dec 31 1862 the 1st Mich Cav received M1859 Spencer Carbines. In Jun 30 1864 the 1st Mich Cav Rgt started to be issued M1860 Spencer Carbines. He also lists type of sabers and pistols issued if you are interested. The information takes several pages in the book so get back with me if you need detailed information.
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