  Classification of AS-regular algebras is one of the main interests in
non-commutative algebraic geometry. Recently, a complete list of
superpotentials (defining relations) of all $3$-dimensional AS-regular algebras
which are Calabi-Yau was given by Mori-Smith (the quadratic case) and
Mori-Ueyama (the cubic case), however, no complete list of defining relations
of all $3$-dimensional AS-regular algebras has not appeared in the literature.
In this paper, we give all possible defining relations of $3$-dimensional
quadratic AS-regular algebras. Moreover, we classify them up to isomorphism and
up to graded Morita equivalence in terms of their defining relations in the
case that their point schemes are not elliptic curves. In the case that their
point schemes are elliptic curves, we give conditions for isomorphism and
graded Morita equivalence in terms of geometric data.
