Which scenario will be worse for the ultimate fate of the Universe, forever expanding or the "Big Crunch"? Will the Universe be destroyed in both scenarios? At the present time does it appear that it will expand forever or will the expansion stop sometime in the future?
Which scenario will be worse for the ultimate fate of the universe?
Well, that's a wee bit opinionated. It depends on your point of view - that is, how you take a liking to the following two scenarios. Everyone has their own opinion; from the details I give below, I'll let you answer that yourself.
To infinity - and . . . well, more infinity: The perpetually expanding universe
There are actually three types of universes that could fit this model: The accelerating expansion one, the constant expansion one, and the decelerating expansion one. At this point in time, we appear to be in a universe undergoing accelerated expansion, thanks to dark energy. However, all three scenarios give us a similar end result: A dead universe.
In the far future, as long as the universe does not curl back into a Big Crunch, slowly all processes as we know them will begin to end. Star formation will eventually cease, and black holes will slowly start to gobble up a lot of matter. In the even farther future, black holes are evaporate to decay via Hawking radiation, leaving the universe a collection of various subatomic particles. (If protons decay, then perhaps there will no longer be any atoms). The temperature of the universe will drop until everything is about thermodynamically equivalent. Not a great place to be (well, there won't be any life, so nobody will be around to see it, but still. . .).
Beyond this. . . well, there are a few different ideas regarding what could happen. The Big Freeze idea states that the universe will eventually just cool off to this low-temperature state. The Big Rip idea says that matter itself could be torn apart bye extreme expansion of space. Cosmological false vacuum theories suggest that quantum tunneling effects could make some interesting expansion possible. However, I emphasize that we do not know whether any of these long-term theories are correct. They are too far in the future, and there is too much uncertainty among the relevant factors.
In the short term, however, it appears our universe will expand forever.
Back to the beginning: The Big Crunch universe and all its friends
The Big Crunch scenario says that if the density of the universe is greater than the critical density, the universe will fall back onto itself into a singularity - just like the Big Bang. This leads to other theories, such as the epykrotic universe, or a Big Bounce, which postulate that our current universe is just undergoing one of infinitely many stages of expansion and collapse.
This isn't exactly a great outcome, either, but again, I'll leave you to judge.
I'll be brief in my answer but I suggest you look at the wikipedia page on dark energy for a good starting point for further research. (http://en.wikipedia.org/wiki/Dark_energy)
At the present time it is believed that the universe will continue to expand at an ever increasing rate due to the driving force of "dark energy" (which is as mysterious as it sounds!). Since space itself is expanding there is no limit to the rate of change in the distance between them, and it is quite possible this speed exceed that of light - meaning observers at each point will not be able to see each other.
With this in mind, eventually (in many billions of years) the universe will be expanding so fast that we will not be able to see any other galaxies, and eventually any other stars after much more time has passed. If the acceleration continues at the increasing rate Earth and the Sun will be "pulled" apart, and at this point the situation on Earth is pretty bleak - no Sun, no stars and no external light.
In the extremely far future it is possible the acceleration of space will cause Earth itself to be torn apart, and the eventual fate of the universe is for every molecule, atom and particle to be broken apart until all that remains are stray photons.
So that's a pretty bad scenario as far as the ultimate fate of the universe goes, but at the moment it seems like that is what may happen.
It's very important to note that extending the increasing rate of acceleration into the future that far is not good practice and probably won't provide very accurate predictions.