  Several new estimates for the 2-adic valuations of Stirling numbers of the
second kind are proved. These estimates, together with criteria for when they
are sharp, lead to improvements in several known theorems and their proofs, as
well as to new theorems. The estimates and criteria all depend on our previous
analysis of powers of 2 in the denominators of coefficients of higher order
Bernoulli polynomials. The corresponding estimates for Stirling numbers of the
first kind are also proved.
  Some attention is given to asymptotic cases, which will be further explored
in subsequent publications.
