I would like to know some examples of the following polar sets (if they exist):

• a non trivial uncountable polar set in $\mathbb{R}^{2}$;
• a polar set in $\mathbb{R}^{2}$ contained in $\mathbb{R}$ with positive Lebesgue measure over $\mathbb{R}$.

I recall that a set $E\subset \mathbb{R}^{2}$ is called polar if there exists a subharmonic function $u:\mathbb{R}^{2}\to \mathbb{R}\cup \{-\infty\}$ such that $E\subset \{u=-\infty\}$.