Let S(n, k) be the clasical Stirling numbers of the second kind, d>1 be an integer, and  P, Q, R in Q[X_1, ..., X_m] be nonconstant polynomials such that P does not divide Q and R is not a d-th power. We prove that if k_1, ..., k_m are any sufficiently large distinct positive integers then, setting S_i = S(n, k_i), Q(S_1, ..., S_m)/P(S_1, ..., S_m) in Z for only finitely many n in N and R(S_1, ..., S_m) = x^d for only finitely many pairs (n, x) in N^2. We extend the latter finiteness result to all triples (n, x, d) in N^3 , d > 1. Our proofs are based on the results of Corvaja and  Zannier. We give similar but more particular results on the more general Stirling-like numbers T(n, k).