  A covariant phase space observable is uniquely characterized by a positive
operator of trace one and, in turn, by the Fourier-Weyl transform of this
operator. We study three properties of such observables, and characterize them
in terms of the zero set of this transform. The first is informational
completeness, for which it is necessary and sufficient that the zero set has
dense complement. The second is a version of informational completeness for the
Hilbert-Schmidt class, equivalent to the zero set being of measure zero, and
the third, known as regularity, is equivalent to the zero set being empty. We
give examples demonstrating that all three conditions are distinct. The three
conditions are the special cases for $p=1,2,\infty$ of a more general notion of
$p$-regularity defined as the norm density of the span of translates of the
operator in the Schatten-$p$ class. We show that the relation between zero sets
and $p$-regularity can be mapped completely to the corresponding relation for
functions in classical harmonic analysis.
