In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\mathbb{Z}$. Let $α(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\mathbb{E}(α(n))$, show that $n\to \mathbb{E}(α(n))$ is non-decreasing in $n$ and that $\lim\limits_{n\to+\infty}\mathbb{E}(α(n))=2$. Then we study the asymptotic behavior of $\mathbb{P} (α(n)>a(\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\in(1/32,1/2]$ such that $\limsup\limits_{n\to+\infty}\frac{α(n)}{(\log n)^2}=C$ almost surely.