(1) Take $x_{0}=2$, then we have
$$f(2)=-26, f(-2)=34 \text {, }$$

Since $f(-2) \neq-f(2)$, and $f(-2) \neq f(2)$,
$\therefore y=x+x^{2}-x^{5}$ is neither an odd nor an even function.
(2) Take $x_{0}=\frac{\pi}{4}$, then we have
$$\begin{array}{l} \text { F }\binom{\pi}{4}=\sin \frac{\pi}{4}-\cos \frac{\pi}{4}=0 \\ \mathrm{f}\left(-\frac{\pi}{4}\right)=\sin \left(-\frac{\pi}{4}\right)-\cos \left(-\frac{\pi}{4}\right) \\ =-\sqrt{2}, \end{array}$$

Since $\mathrm{f}\left(-\frac{\pi}{4}\right) \neq-\mathrm{f}\left(\frac{\pi}{4}\right)$, and
$$\mathrm{f}\left(-\frac{\pi}{4}\right) \neq \mathrm{f}\binom{\pi}{4},$$
$\therefore y=\sin x-\cos x$ is neither an odd nor an even function.