My Hero Academia, Vol. 1 by Kohei Horikoshi

The first foray into a world where eighty percent of the population has superpowers, all unique and so known as Quirks, and one boy without one still passionately desires to be a hero.

Through many mishaps, he does manage to end up at the school. This involves a secret about one hero's quirks, another secret about the rules of the test, a rivalry with another student from his school, a charming girl who can keep him from tripping, a rigorous homeroom teacher, and more.