Let n and k be positive integers, ð(k) and ρ2(k) denote the number of ones in the binary representation of k and the highest power of two dividing k, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that ν2(S(2n,k))=ð(k)-1, 1≤k≤2n. Here we prove that ν2(S(c2n,k))=ð(k)-1, 1≤k≤2n, for any positive integer c. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on ρ2\!(S(c 2n+1+u,k)-S(c2n+u,k)) for any nonnegative integer u, make a conjecture on the exact order and, for u=0, prove part of it when k≤6, or k≥5 and d(k)≤2. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.