  We consider Gaussian random monochromatic waves $u$ on the plane depending on
a real parameter $s$ that is directly related to the regularity of its Fourier
transform. Specifically, the Fourier transform of $u$ is $f\,d\sigma$, where
$d\sigma$ is the Hausdorff measure on the unit circle and the density $f$ is a
function on the circle that, roughly speaking, has exactly $s-\frac12$
derivatives in $L^2$ almost surely. When $s=0$, one recovers the classical
setting for random waves with a translation-invariant covariance-kernel. The
main thrust of this paper is to explore the connection between the regularity
parameter $s$ and the asymptotic behavior of the number $N(\nabla u,R)$ of
critical points that are contained in the disk of radius $R\gg1$. More
precisely, we show that the expectation $\mathbb{E}N(\nabla u,R)$ grows like
the area of the disk when the regularity is low enough ($s<\frac32$) and like
the diameter when the regularity is high enough ($s>\frac52$), and that the
corresponding exponent changes according to a linear interpolation law in the
intermediate regime. The transitions occurring at the endpoint cases involve
the square root of the logarithm of the radius. Interestingly, the highest
asymptotic growth rate occurs only in the classical translation-invariant
setting, $s=0$. A key step of the proof of this result is the obtention of
precise asymptotic expansions for certain Neumann series of Bessel functions.
When the regularity parameter is $s>5$, we show that in fact $N(\nabla u,R)$
grows like the diameter with probability 1, albeit the ratio is not a universal
constant but a random variable.
