  Planck is expected to answer crucial questions on the early Universe, but it
also provides further understanding on anomalous microwave emission. Electric
dipole emission from spinning dust grains continues to be the favored
interpretation of anomalous microwave emission. In this paper, we present a
method to calculate the rotational emission from small grains of irregular
shape with moments of inertia $I_{1}> I_{2}> I_{3}$. We show that a torque-free
rotating irregular grain with a given angular momentum radiates at multiple
frequency modes. The resulting spinning dust spectrum has peak frequency and
emissivity increasing with the degree of grain shape irregularity, which is
defined by $I_{1}:I_{2}:I_{3}$. We discuss how the orientation of dipole moment
$\bmu$ in body coordinates affects the spinning dust spectrum for different
regimes of internal thermal fluctuations. We show that the spinning dust
emissivity for the case of strong thermal fluctuations is less sensitive to the
orientation of $\bmu$ than in the case of weak thermal fluctuations. We
calculate spinning dust spectra for a range of gas density and dipole moment.
The effect of compressible turbulence on spinning dust emission intensity is
investigated. We show that the emission intensity in a turbulent medium
increases by a factor from 1.2-1.4 relative to that in a uniform medium, as
sonic Mach number $M_{s}$ increases from 2-7. Finally, spinning dust parameters
are constrained by fitting our improved model to five-year {\it Wilkinson
Microwave Anisotropy Probe} cross-correlation foreground spectra, for both the
H$\alpha$-correlated and 100 $\mu$m-correlated emission spectra.
