  We have made a systematic numerical study of the 16 Wilf classes of length-5
classical pattern-avoiding permutations from their generating function
coefficients. We have extended the number of known coefficients in fourteen of
the sixteen classes. Careful analysis, including sequence extension, has
allowed us to estimate the growth constant of all classes, and in some cases to
estimate the sub-dominant power-law term associated with the exponential
growth.
  In six of the sixteen classes we find the familiar power-law behaviour, so
that the coefficients behave like $s_n \sim C \cdot \mu^n \cdot n^g,$ while in
the remaining ten cases we find a stretched exponential as the most likely
sub-dominant term, so that the coefficients behave like $s_n \sim C \cdot \mu^n
\cdot \mu_1^{n^\sigma} \cdot n^g,$ where $0 < \sigma < 1.$ We have also
classified the 120 possible permutations into the 16 distinct classes.
  We give compelling numerical evidence, and in one case a proof, that all 16
Wilf-class generating function coefficients can be represented as moments of a
non-negative measure on $[0,\infty).$ Such sequences are known as {\em
Stieltjes moment sequences}. They have a number of nice properties, such as
log-convexity, which can be used to provide quite strong rigorous lower bounds.
  Stronger bounds still can be established under plausible monotonicity
assumptions about the terms in the continued-fraction expansion of the
generating functions implied by the Stieltjes property. In this way we provide
strong (non-rigorous) lower bounds to the growth constants, which are sometimes
within a few percent of the exact value.
