  Let $H = \Delta^2 + V$ be the fourth-order Schr\"odinger operator on
$\mathbb{R}^3$ with a real-valued fast-decaying potential $V$. If zero is
neither a resonance nor an eigenvalue of $H$, it was recently known that the
wave operators $W_\pm(H, \Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all
$1 < p < \infty$ and unbounded at the endpoints $p=1$ and $p=\infty$.
  This paper is to further establish the $L^p$-bounds of $W_\pm(H, \Delta^2)$
that exhibit all types of singularities at the zero energy threshold. We first
prove that $W_\pm(H, \Delta^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1 <
p < \infty$ in the first kind resonance case, and then proceed to establish for
the second kind resonance case that they are bounded on $L^p(\mathbb{R}^3)$ for
all $1 < p < 3$, but not for any $3 \le p \le \infty$.
  In the third kind resonance case, we still show that $W_\pm(H, \Delta^2)$ are
bounded on $L^p(\mathbb{R}^3)$ for all $1<p<3$ and generally unbounded on
$L^p(\R^3)$ for any $3\le p\le\infty$. However, it can be also shown that
$W_\pm(H, \Delta^2)$ remain bounded on $L^p(\R^3)$ for all $3\le p<\infty$ if
in addition $H$ has a zero eigenvalue, but no $p$-wave zero resonance and all
zero eigenfunctions are orthogonal to $x_ix_jx_kV$ in $L^2(\R^3)$ for all
$i,j,k=1,2,3$ with $x=(x_1,x_2,x_3)$.
  These results precisely describle all $L^p$-bounds of the wave operators
$W_\pm(H, \Delta^2)$ in $\mathbb{R}^3$ with some exceptions of the endpoint
cases for $p=1$ and $p=\infty$. As a result, $L^p$-$L^q$ decay estimates are
derived for the fourth-order Schr\"odinger equations and Beam equations with
zero resonance singularities.
