p-vanishing conjugacy classes of symmetric groups

For a prime p, we say that a conjugacy class of a finite group G is p-vanishing if every irreducible character of G of degree divisible by p takes value 0 on that conjugacy class. In this paper we completely classify 2-vanishing and 3-vanishing conjugacy classes for the symmetric group and do some work in the classification of p  -vanishing conjugacy classes of the symmetric group for p≥5p≥5. This answers a question by Navarro for p=2p=2 and p=3p=3 and partly answers it for p≥5p≥5.