1. In 1924, she finally entered flight school.

2. In 1924, she finally entered the flight school.

3. In 1924, she finally entered a flight school.
-----------
Which one is grammatical or commonly used?

All are correct, but they don't all mean exactly the same thing.

1 and 3 mean any flight school, in general.

2 means a particular flight school that both the speaker and the listener know of.