We consider $O$-sequences that occur for arithmetically
Cohen-Macaulay (ACM) schemes $X$ of
codimension three in ${\pp}^n$. These are Hilbert functions
$\varphi$ of Artinian algebras that are quotients of the
coordinate ring of $X$ by a linear system of parameters. Using
suitable decompositions of $\varphi$, we determine the minimal
number of generators possible in some degree $c$ for the
defining ideal of any such ACM scheme having the given
$O$-sequence. We apply this result to construct Artinian
Gorenstein $O$-sequences $\varphi$ of codimension $3$ such
that the poset of all graded Betti sequences of the Artinian
Gorenstein algebras with Hilbert function $\varphi$ admits
more than one minimal element. Finally, for all
$3$-codimensional complete intersection $O$-sequences we
obtain conditions under which the corresponding poset of
graded Betti sequences has more than one minimal element.