Michigan's best streak was obviously before OSU joined the Big Ten.
They won the first game in 1897, which remains OSU's worst season in school history.
They played every year from 1900-1912.
The first game in that stretch ended in a tie. Michigan finished 4th in the Big Ten, and OSU split a "State Championship" with two other teams.
Fielding Yost was hired in 1901. Michigan won that game en route to an undefeated Big Ten season capped with a win in the first Rose Bowl.
From 1902-06 Michigan was in the Big Ten and OSU was in the newly formed OAC. Michigan swept OSU during that stretch, and won the Big Ten all but one year. OSU won an OAC Title in 06, but lost to Michigan 6-0.
From 1907-12 Michigan was Independent while OSU remained in the OAC. Michigan went 5-0-1 vs OSU over that stretch, including a 14-0 win over OSU's 1912 OAC Title team. The lone tie was in 1910 where Michigan went undefeated, albeit with 3 wins and 3 ties.
So Michigan was 12-0-2 over OSU before OSU joined the Big Ten.
The series went on hiatus from 1913-1917, as a condition of OSU's admission into the Big Ten.
Michigan won the first "Big Ten" game between the two schools in 1918, during Chic Harley's one year hiatus from football. Harley returned to OSU in 1919 for his senior year, and led OSU to their first victory over the Wolverines in school history; kicking off a three game winning streak for the good guys.