To which Sobolev local space Dirac delta function belongs to?

To which Sobolev local space Dirac delta function belongs to?

I have found that Dirac delta function $\delta (x)\in H^{s}(\mathbb{R}),
\forall s<-\frac{1}{2}$, and Heaviside function $\in H^{s}(\mathbb{R}) ,
\forall s<\frac{1}{2}$;
Also i want to use the following formula for Cauchy principal value
$p.v\left(\frac{1}{x}\right)$: $\hat{H}(\xi)=\frac{1}{2}\left (
\delta(\xi)-\frac{i}{\pi}p.v\left(\frac{1}{\xi}\right) \right )$ in order
to find to which $H^{s}(\mathbb{R})$ does $p.v\left(\frac{1}{x}\right)$
belong?
How can I derive it using this formula? Or what is another way to find to
which $H^{s}(\mathbb{R})$ does $p.v\left(\frac{1}{x}\right)$ belong?
Thanks a lot.