  In this paper, we look at the question of when various ideals in the Fourier
algebra $A(G)$ or its closures $A_M(G)$ and $A_{cb}(G)$ in, respectively, its
multiplier and $cb$-multiplier algebra are Arens regular. We show that in each
case, if a non-zero ideal is Arens regular, then the underlying group $G$ must
be discrete. In addition, we show that if an ideal $I$ in $A(G)$ has a bounded
approximate identity, then it is Arens regular if and only if it is
finite-dimensional.
