Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$. Its Weyl group is the symmetric group $S_n$. In this paper, we want to describe some Kazhdan-Lusztig right cells containing smooth elements which parameterize the smooth Schubert varieties. These elements are closely related to the study of associated varieties of highest weight modules of $\mathfrak{sl}(n,\mathbb{C})$. Firstly, we give a complete classification of the KL right cells containing only smooth elements. Then we give a sufficient condition for a KL right cell to contain only non-smooth elements by using invariant subsequences and a sufficient condition for a KL right cell to contain some smooth elements. Finally, we give an efficient algorithm to find out all the smooth elements in a given KL right cell.