Do all black holes have a singularity?

Short answer is yes.

But if you want to nit pick, I could argue that when a star collapses to form a BH, it first forms a horizon before the singularity forms (cannot form a "naked singularity"). And since time inside the horizon is essentially frozen with respect to that of an observer outside, the singularity NEVER forms. Yet from the point of view of the collapsing star, the singularity forms in about a millisecond after the horizon.

In classical General Relativity, once an event horizon forms, every particle inside that event horizon will inevitably travel in the direction of the center of the black hole. This is what is meant by "gravitational collapse" and how matter comes to form a singularity in the center- no matter how small it is, or how close to the center it is, nothing can prevent it from approaching ever closer to the center. From the point of view of the object itself, it does reach the center in a finite time.

In some more exotic theories of physics, such as string theory or loop quantum gravity, the quantized nature of space and time comes to the rescue and prevents a singularity from forming, so a maximum, finite density is reached and an equilibrium is maintained in the center. This is similar conceptually to what you describe, but still a more exotic and much, much denser material than neutron star-stuff.

The density we're talking about here would be approximately one Planck Mass per cubic Planck Length, in other words 2.176 51 × 10^−8 kg / (1.616 199 × 10^−35 m)^3 ~= 5.15556^96 kg/m^3, where neutron star material is "only" (roughly) 10^18 kg/m^3.

In either case, however, outside the event horizon, the black hole can be treated mathematically and observationally as a simple singularity, so for observational calculations, there is no "value added" in worrying about the inner workings of the black hole. The theorem describing this is colloquially called "Black holes have no hair." This theorem was proven and coined by John Wheeler, the same physicist who coined the phrase "black hole" in the first place.

Wikipedia seems to indicate they they all do:

At the center of a black hole as described by general relativity lies a gravitational singularity, a region where the spacetime curvature becomes infinite. For a non-rotating black hole this region takes the shape of a single point and for a rotating black hole it is smeared out to form a ring singularity lying in the plane of rotation. In both cases the singular region has zero volume. It can also be shown that the singular region contains all the mass of the black hole solution. The singular region can thus be thought of as having infinite density.