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We know that the $P$ function of (displaced) thermal states is smooth and positive. Indeed, as shown e.g. here, a thermal state $\rho_{\rm th}(x)\equiv (1-x)x^{a^\dagger a}$, $x\equiv e^{-\beta}$, has $P$ function $$P_x(\nu) = \frac{1-x}{\pi x} \exp\left(-\frac{1-x}{x} |\nu|^2\right).$$ Displaced thermal states of course give essentially the same expression. In the limit $x\to0$ this becomes $\delta^2(\nu)$, consistently with the thermal state approaching the vacuum state $|0\rangle$.

Other typical examples of states with known $P$ function are Fock states: the $P$ function of the Fock state $|n\rangle$ is a $2n$-th derivative of $\delta^2(\nu)$ (and is thus highly singular).

What are other "simple" examples of states that have a non-singular $P$-function? Of course, any mixture of (displaced) thermal states would foot the bill, but I'm looking for other nontrivial examples. For example, any example involving a pure state would be nice. Ideally, cases where the $P$ function can be computed analytically (or at least shown analytically to be regular).

glS
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Hudson's theorem established that all pure states with positive Wigner functions are Gaussian states (i.e., they are squeezed and/or displaced vacuum states, perhaps multimode versions). Since the Wigner function is a convolution of the P function, the negativity of the Wigner function implies the negativity of the P function, so the only possible states with positive P functions are those with positive Wigner functions.

Thus, if you really want pure states with positive P functions, the only possibilities are squeezed coherent states. Then you simply show that squeezing always makes a state have a negative P function (any squeezed state has negative P somewhere, displacements only move that around in phase space) and you discover that the only pure states with positive P distributions are coherent states, there are no others.

Quantum Mechanic
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