To make this question more precise, we need to define terms a little better. The standard definition of a black hole is the following. Suppose there are points in spacetime from which it's impossible to escape to a large distance. (Technically, we want these to be points from which we can't escape to future null infinity.) If we have set of such points, then the boundary of that set is a black hole event horizon. A black hole is defined as a region of space surrounded by this particular type of event horizon. So there is nothing in the definition that directly requires a singularity.
It is certainly possible to have a horizon without a singularity. In fact, horizons are observer-dependent. In flat (Minkowski) spacetime, you can have an observer with constant proper acceleration, and for that observer, there is a horizon. Events behind the horizon can never send a signal that the observer will be able to receive. However, this horizon isn't the boundary of a black-hole event horizon, so there is no black hole.
There are two different theorems in GR that address this question of whether you can have an event horizon without a singularity. They say slightly different things.
The Penrose singularity theorem
There is a concept very similar to that of an event horizon, which is a trapped lightlike surface. This is a surface such that even if you emit light rays from it in the outward direction, the resulting surface formed by the emitted rays has a decreasing volume. If such a trapped surface exists, then the Penrose singularity theorem guarantees that the spacetime contains a singularity.
This theorem is important because although we know that there is a limit on the mass of a stable neutron star (the Tolman-Oppenheimer-Volkoff limit), such limits assume static equilibrium. In a dynamical system like a globular cluster, the generic situation in Newtonian gravity is that things don't collapse in the center. They tend to swing past, the same way a comet swings past the sun, and in fact there is an angular momentum barrier that makes collapse to a point impossible. The Penrose singularity theorem tells us that general relativity behaves qualitatively differently from Newtonian gravity for strong gravitational fields, and collapse to a singularity is in some sense a generic outcome. The singularity theorem also tells us that we can't just keep on discovering more and more dense forms of stable matter; beyond a certain density, a trapped lightlike surface forms, and then it's guaranteed to form a singularity.
No-hair theorems
A different type of theorem relates more directly to event horizons. These are the black hole no-hair theorems, which apply assuming that the resulting system settles down at some point (technically the assumption is that the spacetime is stationary). Basically, the no-hair theorems say that if an object has a certain type of event horizon, and if it's settled down, it has to be a black hole, and can differ from other black holes in only three ways: its mass, angular momentum, and electric charge. These well-classified types all have singularities.
Of course these theorems are proved within general relativity. In a theory of quantum gravity, probably something else happens when the collapse reaches the Planck scale.
Observationally, we see objects such as Sagittarius A* that don't emit their own light, have big masses, and are far too compact to be any stable form of matter with that mass. This strongly supports the validity of the above calculations and theorems. Even stronger support will come if we can directly image Sagittarius A* with enough magnification to resolve its event horizon. This may happen within 10 years or so.
For a more in-depth discussion of this sort of thing, see Booth, http://arxiv.org/abs/gr-qc/0508107
