Polynomials with prescribed bad primes

We tabulate polynomials in Z[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and infinity. We explain how these sets of polynomials are of particular interest because of their role in the construction of nonsolvable number fields of arbitrarily large degree and bounded ramification. Finally we discuss the similar but technically more complicated tabulation problem corresponding to removing the auxiliary conditions.