Does general relativity entail singularities if there's a positive cosmological constant? I've heard that Hawking and Penrose proved that general relativity entails singularities.  But it says in the abstract of what seems to be the paper in which they proved it (The Singularities of Gravitational Collapse and Cosmology) that the theorem applies only if certain assumptions are made, one of which is a zero or negative cosmological constant.
Hasn't a positive cosmological constant been favored since the 1998 discovery that the expansion of the universe is accelerating?  If so, is it known (i.e. mathematically proven or clearly established from physical evidence) that general relativity entails singularities in that case?   
 A: I would say the sign of the cosmological constant would certainly play a factor in determining singularity behaviour of the universe. This can be seen from Raychaudhuri’s equation, which is precisely obtained from Einstein’s field equations, and is given by:
$$\dot{\theta} + \frac{1}{3} \theta^2 + \sigma_{uv}\sigma^{uv} - \omega_{uv} \omega^{uv} + \frac{\kappa}{2} (\mu + 3p) - \Lambda = 0$$
where $\theta$ is the expansion scalar, $\sigma_{uv}$ is the shear tensor, $\omega_{uv}$ is the vorticity, $\mu$ is the energy density, $p$ is the pressure, and $\Lambda$ is the cosmological constant. (This is not the most general form of the Raychaudhuri equation, as I have assumed that the universe model is spatially homogeneous, which has simplified things quite a bit (all partial derivatives are now ordinary time derivatives, however, it will illuminate this discussion slightly). Also, the Raychaudhuri equation was the main motivation behind the Penrose-Hawking singularity theorems.
Now, our universe is understood to be spatially homogeneous and isotropic on the largest scales, and as such, by these symmetries, we must have that the shear and vorticity vanish, such that Raychaudhuri’s equation becomes:
$$\dot{\theta} + \frac{1}{3} \theta^2 + \frac{\kappa}{2} (\mu + 3p) - \Lambda = 0$$
There are many ways to get $\theta(t) \to \infty$, and they depend on the curvature of the universe, the sign of the cosmological constant, the pressure/energy density in the universe, the nature of dark energy,etc… Many models exist in the scientific literature which discuss these issues at length. For example, the recollapse theorems of Barrow and Tipler are actually much more general than the Penrose-Hawking singularity theorems, as Barrow and Tipler use the full Einstein equations, while Penrose-Hawking limit their studies to timelike geodesics.
Barrow/Tipler Paper 
A: A singularity involves an infinite amount of negative potential energy in a localized volume.  A nonzero cosmological constant would only yield a finite amount of positive energy in a localized volume.  So the cosmological constant might slow down the rate of singularity production, but it won't stop it.
A: Singularities are very likely impossible to create in the real universe. 
In other words, as a singularity gets close to forming, the incoming random GR waves and other energy will rip the formation apart, keeping it in a state of almost singularity. 
As an example, all Black Holes spin in the real world. The size of the singularity in a spinning Kerr geometry is within a hair of being zero:

Thus we reach the conclusion that at timeline or null geodesic or
  orbit cannot   reach the singularity under any circumstances except in
  the case where it is confined to the equator, cos() = 0…..Thus as
  symmetry is progressively reduced, starting from the Schwarchild
  solution, the extent of the class of geodesics reaching the
  singularity is steadily reduced likewise, … which suggests that after
  further reduction in symmetry, incomplete geodesics may cease to exist
  altogether
Kerr Fields, Brandon Carter 1968.

So while General Relativity in theory has singularities, its not likely that any exist in a real noisy universe. The cosmological constant does not I think enter into the problem.
The wikipedia page on the singularity theorems says as much as well. 
https://en.wikipedia.org/wiki/Penrose–Hawking_singularity_theorems

It is still an open question whether time-like singularities ever
  occur…

