  We study various measures of classicality of the states of open quantum
systems subject to decoherence. Classical states are expected to be stable in
spite of decoherence, and are thought to leave conspicuous imprints on the
environment. Here these expected features of environment-induced superselection
(einselection) are quantified using four different criteria: predictability
sieve (which selects states that produce least entropy), purification time
(which looks for states that are the easiest to find out from the imprint they
leave on the environment), efficiency threshold (which finds states that can be
deduced from measurements on a smallest fraction of the environment), and
purity loss time (that looks for states for which it takes the longest to lose
a set fraction of their initial purity). We show that when pointer states --
the most predictable states of an open quantum system selected by the
predictability sieve -- are well defined, all four criteria agree that they are
indeed the most classical states. We illustrate this with two examples: an
underdamped harmonic oscillator, for which coherent states are unanimously
chosen by all criteria, and a free particle undergoing quantum Brownian motion,
for which most criteria select almost identical Gaussian states (although, in
this case, predictability sieve does not select well defined pointer states.)
