3 many a (formal) used with a singular noun and verb to mean ‘a large number of’:
Many a good man has been destroyed by drink.
Student or Learner
Many a day finds me locked in my room, studying
Many a day find me locked in my room, studying
I am not a teacher, but I am a native speaker.
"Many a day" is somewhat of an old phrase. The vast majority of native English speakers do not use it anymore, and it sounds kind of odd. However, it would be: "Many a day find me locked in my room studying".
A native speaker would be much more likely to say it: "Most days find me locked in my room studying." But the other way is correct English. It is just much less common.
I agree with bhaisahab.