  In this paper we study Hilbert functions and isomorphism classes of Artinian
level local algebras via Macaulay's inverse system. Upper and lower bounds
concerning numerical functions admissible for level algebras of fixed type and
socle degree are known. For each value in this range we exhibit a level local
algebra with that Hilbert function, provided that the socle degree is at most
three. Furthermore we prove that level local algebras of socle degree three and
maximal Hilbert function are graded. In the graded case the extremal strata
have been parametrized by Cho and Iarrobino.
