Many a...plural or singular?

No no no, it's always singular. Your own examples of "many a time", "many a true word" show singular nouns, and it's the same with "many a book study", "many a bar downtown". Time, word, study and bar are all nouns. You have no option - whatever YOU "deem fit", the noun must always be singular. The clue lies in the word "a" - which by definition is singular.