Does this implies that a spaceship, inside a black hole, accelerating in the direction OPPOSITE to the singularity would actually accelerate TOWARDS the singularity? Or does it simply mean that no matter how fast a speed he reaches through his acceleration, he would never get to a speed fast enough to stop his falling towards the center?

Also, is it true that an object inside a black hole will never quite reach the center? That would make a lot of sense for a black hole consisting of an infinitely dense star. However, I think some (most?) people on this forum would tend to believe the star doesn't collapse infinitely on itself but rather has a minimum size (something I would intuitively tend to believe too). In this scenario, it would seem that matter falling in a black hole would be able to reach the center after some time. Correct?

Lastly, if the center of the BH is NOT an infinitely dense star but rather has some definite volume, would it still be called a 'singularity' or is the word stritcly reserved for points at which the curvature of space-time is infinite?

The term "singularity" is indeed reserved for a mass with no dimensions, occupying a single point.

I am one of those who do not believe BH's have a singularity at the center, and your statement about time-dilation making it take forever to reach the center is precisely the reason why. If time dilation makes the final moment of infall take forever, then this also applies the final moment of collapse for the original mass. So theoretically, niether the original mass nor any infalling object would ever reach the center.

I was under the impression that pretty much no scientists though there was a singularity in a black hole anymore? I thought the only possible singularity anyone talked about seriously now was pre-big bang? But maybe I'm wrong.

Since when? An infinite 2-dimentional line of mass-energy (e.g. a beam of light with no width) has a 2d-singularity. It doesn't even have an event horizon. I guess that's called a naked singularity.

Nothing can accelerate in the direction opposite to the singularity.

And I don't believe that anything can reach the singularity since it would take an infinite amout of time to do so. I think the distance from a point outside the BH to the singularity is infinite - Don't quote me though since I'm not really into black holes.

Acceleration, in whatever direction, will only mean that it reaches the singularity sooner. The way to survive longest is to stay in free fall. I'm not sure that is makes sense to talk of accelerating in the direction OPPOSITE to the singularity as this direction is now timelike - how do you accelerate backwards in time?

Using Schwarzchild coordinates (the coordinate system of an outside observer) an object takes an infinite amount of time to reach the event horizon and even longer to reach the singularity. However from the point of view of the infalling obect it crosses the event horizon and reaches the singularity within a finite (and for typical black holes, fairly short) time.

Quantum theory suggests that there is some limit to how small the collapsing star can become. However we don't have a quantum theory of gravity yet so its hard to know. My guess is that black holes will turn out not to exist (see www.chronon.org/articles/blackholes.html)

I would think that full quantum gravity would also eliminate the big-bang singularity

Although no mass reaches the center in finite time, the gravitational effects of the mass, as seen from outside the event horizon, are indistinguishable from the gravitational effects of a point singularity. If you could magically teleport to the center of a black hole, without waiting forever to get there, you would see empty space and null gravity. But, without magic, there's no way to reach the center of mass ahead of the mass that's already gotten "sufficiently close" to the event horizon.
If you free-fall to the center, you get there in finite time in your worldline, but the mass needed to form the actual singularity gets there ahead of you (or at the same time, at the end of time).

I have even seen an article explaining how, within the event Horizon, even centrifugal force works in reverse. A traveler attempting to keep his distance from the center of black hole by accelerating laterally and trying to "orbit" the center will find that, the faster he orbits, the more force he generates toward the center! I found this completely mind blowing.