  We prove the rank of the group of signatures of the circular units (hence
also the full group of units) of ${\mathbb Q}( \zeta_m)^+$ tends to infinity
with $m$. We also show the signature rank of the units differs from its maximum
possible value by a bounded amount for all the real subfields of the composite
of an abelian field with finitely many odd prime-power cyclotomic towers. In
particular, for any prime $p$ the signature rank of the units of ${\mathbb Q}(
\zeta_{p^n})^+$ differs from $\varphi(p^n)/2$ by an amount that is bounded
independent of $n$. Finally, we show conditionally that for general cyclotomic
fields the unit signature rank can differ from its maximum possible value by an
arbitrarily large amount.
