Are galactic stars spiraling inwards? Are the stars in our galaxy spiraling inwards towards the center, or are they in a permanent orbit?
And if we are heading towards the center then what is the rate of this process?
I started wondering this while watching this documentary: http://www.youtube.com/watch?v=zKE4Bt8ylhM
 A: All stars in our galaxy are in stable elliptical orbits around the galactic centre. But they are not all moving in the same direction with the same speed... meaning there is a random maxwellian distribution of velocities among the stars. 
What this effectively means is (like the animation that Crazy Buddy posted) although there is a net effective attraction between the stars and the centre of the galaxy, whenever two stars get close to each other, they exert a sort of "gravitational drag" force, which acts like a frictional force and ends in a net slowing down of the interacting bodies. 
When the star then slows down, it doesn't have the speed to maintain it's current orbit, so it will move into a nearer orbit closer to the centre of the galaxy. This is the general mechanism of how stars in a galaxy are "collapsing" into the centre. 
There's no need to worry about falling into any (hypothesised) super massive black holes though. Any of these processes take a really long time, and by the time we are close enough to the centre of the galaxy to worry about it, our sun will be well into it's red giant stage and the earth would already have been consumed by it. :)
P.S - To clarify my first paragraph, assuming there are no other stars orbiting the centre of the galaxy, then we will never spiral into it. That effect is only caused by the presence of other stars in the galaxy.
EDIT: @JohnRennie pointed out that because of conservation of momentum, the lighter stars will tend to gain energy and the heavier ones loose energy in dynamical friction interactions. This will tend to push the heavier stars closer to the centre and the lighter stars further out. His answer to this same question points this out. 
A: This is really just a footnote to Kitchi's answer and Hermann's comment to it:
Dynamical friction won't cause all the stars in the galaxy to swirl down into the black hole like water down a plug hole. Apart from anything else conservation of momentum forbids this. Dynamical friction causes a type of sorting. In interactions between stars on average the heavier stars tend to lose energy and the lighter stars tend to gain energy. This will concentrate the heavier stars towards the centre of the galaxy and expel the lighter ones to its edge. In fact lighter stars can be completely ejected from the galaxy - see page 521 of this book for more details.
For stars to merge with the black hole they need to lose a great deal of their energy. The diameter of Sagittarius A* is about 35 light hours while the diameter of the galaxy is about 100,000 light years i.e. the galaxy is about 25 million times bigger than the black hole at it's centre. For a star to lose just the right amount of energy for it's orbit to intersect the black hole is extremely unlikely. It will eventually happen, but i wouldn't hold your breath.
A: The stars are orbiting our Milky Way's center for now. But, perhaps this isn't a Full stop. There are probabilities for these to fall into the center and crush into nowhere. So, yes indeed. We are spiraling inner towards the center. An amazing gravitational mass is thought to exist at the galactic center. As we're at some pretty longer distance, the effect maybe negligible. But, we're (I mean, our solar system) being accelerated spirally towards the center.

An add-up for WHY? : Though most galaxies are thought to have a black-hole at their center, our galaxy indeed has a black-hole (a super-massive one) which has been pretending now & then to be the galactic center. Scientists are a bit wondered by it as the physical proof for Einstein's GR may or may not arrive (because physicists would get happy for both failure and success). What do ya think about the star in the image below. Definitely, it's not a star because other stars are orbiting it :-)
A great proof is Wiki and several other articles based on the discovery which you could Google easily. Have a look at the stars S0-2 and S0-102 which orbit our massive black-hole.
                 
A: Suppose an alone star orbiting a galactic center for simplicity.
It remains on its orbit due to conservation of angular momentum. There are three factors that may cause it to loose its angular momentum


*

*Resistance of interstellar medium

*Tidal forces

*Emitting of gravitational waves
If the center of the galaxy also rotates in the same direction, with greater speed, it will transfer its angular momentum to the orbiting star thus making it to temporally increase the distance.
The first factor is the greatest of them while the others are negligible compared to it. Yet all the three are so small that they even cannot sufficiently affect the planetary orbits in Solar System for billions of years.
It is safe to assume that the whole substance of the star will completely evaporate through such great time span before it collides with the galactic center.
The planets orbiting such star of course will fall on it in a much shorter time.
Yet in the presence of other stars it is much more likely that Sun will collide with other stars in a much closer future, or be ejected from the galaxy completely.
