I have following equation $G(z)={z+0.5\over z^2-1.5z+0.5}$ with Partial fraction decomposition I got $G(z)={3\over z-1}+{-2\over z-0.5}$ usually we look in a conversion table how to do the reverse z-transformation.
unfortunately I cannot find anything suiting my equation. I'd greatly appreciate it if someone could help me! Thank you very much.
If these are actually $z$ transforms (not Laplace transforms), the main thing you want to refer back to is
$$\frac{1}{1-z}=\sum_{n=0}^\infty z^n.$$
Now do some algebra to make each of your expressions look like $\frac{C}{1-u(z)}$ for some constant $C$ and a suitable expression $u(z)$, and then plug into the formula above. (Hint: in the first one $u(z)$ is just $z$.)