  In this work, we show that if $f$ is a uniformly continuous map defined over
a Polish metric space, then the set of $f$-invariant measures with zero metric
entropy is a $G_\delta$ set (in the weak topology). In particular, this set is
generic if the set of $f$-periodic measures is dense in the set of
$f$-invariant measures. This settles a conjecture posed by Sigmund (Sigmund, K.
On dynamical systems with the specification property. Trans. Amer. Math. Soc.
190 (1974), 285-299) which states that the metric entropy of an invariant
measure of a topological dynamical system that satisfies the periodic
specification property is typically zero.
  We also show that if $X$ is compact and if $f$ is an expansive or a Lipschitz
map with a dense set of periodic measures, typically the lower correlation
entropy for $q\in(0,1)$ is equal to zero. Moreover, we show that if $X$ is a
compact metric space and if $f$ is an expanding map with a dense set of
periodic measures, then the set of invariant measures with packing dimension,
upper rate of recurrence and upper quantitative waiting time indicator equal to
zero is residual.
  Finally, we present an alternative proof of the fact that the set of
expansive measures is a $G_{\delta\sigma}$ set in the set of probability
measures $\M(X)$, if $X$ is a Polish metric space and if $f$ is uniformly
continuous; this result was originally proved by Lee, Morales and Shin (Lee,
K., Morales, C. A., and Shin, B. On the set of expansive measures. Commun.
Contemp. Math. 20, 7 (2018), 1750086, 10) for compact metric spaces.
