For a natural number $n$, let $P_n$ denote a Sylow $p$-subgroup of the symmetric group $S_n$. In 2017 E. Giannelli and G. Navarro proved that if $\chi$ is an irreducible character of $S_n$ with degree divisible by $p$, then the restriction of $\chi$ to $P_n$ has at least $p$ different linear constituents. In this talk we will present the result that classifies the set of irreducible characters of the symmetric groups whose restriction to $P_n$ have at most $p$ linear constituents when $p=2$. We will also for mention the multiplicity of these linear characters for certain families of irreducible characters of $S_n$.