The purpose of this thesis is to investigate the structure of the restriction to Sylow subgroups of irreducible characters of the symmetric groups. In particular, given a natural number k, which and how many irreducible characters of a specific symmetric group admit a constituent of degree p^k in their restriction to the Sylow p-subgroup? We answer this question in two different ways for an odd prime and for the prime being equal 2. In the last part, we focus on the modular representation theory of the symmetric groups, describing when a Young module is also a Scott module.