  The striking differences between quantum and classical systems predicate
disruptive quantum technologies. We peruse quantumness from a variety of
viewpoints, concentrating on phase-space formulations because they can be
applied beyond particular symmetry groups. The symmetry-transcending properties
of the Husimi $Q$ function make it our basic tool. In terms of the latter, we
examine quantities such as the Wehrl entropy, inverse participation ratio,
cumulative multipolar distribution, and metrological power, which are linked to
intrinsic properties of any quantum state. We use these quantities to formulate
extremal principles and determine in this way which states are the most and
least "quantum;" the corresponding properties and potential usefulness of each
extremal principle are explored in detail. While the extrema largely coincide
for continuous-variable systems, our analysis of spin systems shows that care
must be taken when applying an extremal principle to new contexts.
