18 D.
$$
\begin{array}{c}
\ln \left(-x+\sqrt{x^{2}+1}\right)^{4}=\ln \left(\frac{1}{x+\sqrt{x^{2}+1}}\right)^{4}=-\ln \left(x+\sqrt{x^{2}+1}\right)^{4} . \\
\frac{1+\sin (-2 x)+\cos (-2 x)}{1+\sin (-2 x)-\cos (-2 x)}=\frac{1-\sin 2 x+\cos 2 x}{1-\sin 2 x-\cos 2 x}
\end{array}
$$

And
$$
\begin{aligned}
& (1+\sin 2 x+\cos 2 x)(\sin 2 x+\cos 2 x-1) \\
= & (\sin 2 x+\cos 2 x)^{2}-1 \\
= & 2 \sin 2 x \cos 2 x, \\
& (1+\sin 2 x-\cos 2 x)(1-\sin 2 x+\cos 2 x) \\
= & 1-(\sin 2 x-\cos 2 x)^{2} \\
= & 2 \sin 2 x \cos 2 x,
\end{aligned}
$$

Therefore,
$$
\begin{array}{l}
\frac{1+\sin (-2 x)+\cos (-2 x)}{1+\sin (-2 x)-\cos (-2 x)}=-\frac{1+\sin 2 x+\cos 2 x}{1+\sin 2 x-\cos 2 x} . \\
\frac{1+2 \sin \frac{-x}{2}+\sin (-x)-\cos (-x)}{1+2 \cos \frac{-x}{2}+\sin (-x)+\cos (-x)}=\frac{1-2 \sin \frac{x}{2}-\sin x-\cos x}{1+2 \cos \frac{x}{2}-\sin x+\cos x}, \\
\left(1+2 \cos \frac{x}{2}-\sin x+\cos x\right)\left(1+2 \sin \frac{x}{2}-\sin x-\cos x\right)+ \\
\left(1+2 \cos \frac{x}{2}+\sin x+\cos x\right)\left(1-2 \sin \frac{x}{2}-\sin x-\cos x\right) \\
=2\left(1+2 \cos \frac{x}{2}+\cos x\right)(1-\cos x)-2 \sin x\left(2 \sin \frac{x}{2}+\sin x\right) \\
=4\left(\cos \frac{x}{2}-\cos x \cos \frac{x}{2}-\sin x \sin \frac{x}{2}\right) \\
=4\left(\cos \frac{x}{2}-\cos \frac{x}{2}\right)=0, \\
\end{array}
$$

Therefore,
$$
\begin{array}{c}
\frac{1-2 \sin \frac{-x}{2}+\sin (-x)-\cos (-x)}{1+2 \cos \frac{-x}{2}+\sin (-x)+\cos (-x)}=-\frac{1-2 \sin \frac{x}{2}+\sin x-\cos x}{1+2 \cos \frac{x}{2}+\sin x+\cos x} . \\
1-2^{-(-x)}=1-2^{x}=-\left(2^{x}-1\right), \\
2^{-x}-1=-\left(1-2^{-x}\right) . \\
\frac{a(-x)}{(-x)^{2}-1}=-\frac{a x}{x^{2}-1} .
\end{array}
$$

Therefore, all five functions are odd functions.
18 The following functions:
(1) $f(x)=\ln \left(x+\sqrt{x^{2}+1}\right)^{4}$,
(2) $f(x)=\frac{1+\sin 2 x+\cos 2 x}{1+\sin 2 x-\cos 2 x}$,
(3) $f(x)=\frac{1+2 \sin \frac{x}{2}+\sin x-\cos x}{1+2 \cos \frac{x}{2}+\sin x+\cos x}$,
(4) $f(x)=\left\{\begin{array}{ll}1-2^{-x}, & x \geqslant 0, \\ 2^{x}-1, & x<0,\end{array}\right.$
(5) $f(x)=\frac{a x}{x^{2}-1}(a \in \mathbf{R})$,
are not odd functions:
A. 3
B. 2
C. 1
D. 0