Comparing the ultra-wide, crop-sensor camera lenses is an extremely difficult task, so I’ll put the summary first: they all deliver excellent image quality and you can’t go wrong with any of them. To my ‘just taking pictures’ assessment they are all excellent. There are some differences though, so I’ll try to point those out so you have a better chance at choosing the one that’s best for you.
The Nikon 10-24 f/3.5-4.5G Shows a fair amount of barrel distortion at 10mm (note to self: never take girlfriend’s portrait with this lens) and doesn’t have the best corner sharpness of the group. Its got the largest range and is fast to autofocus, which makes it a great wide-angle walk around lens. If I was mostly going to shoot at 15-24 and occasionally shoot wider, this would be a great choice. If I was going to take lots of 10mm scenic shots, I’d probably look at one of the others – they’re a bit better at 10mm.
The Nikon 12-24 f/4G has exceptional center resolution and is very flare resistant, but does show some field curvature at 12mm @ f/4 the border quality is also impressive though a little less field curvature may be desirable at 12mm. Its a very good lens, maybe better than the competition, but not as much better as its price would suggest.
The Tokina 11-16 f/2.8 gives you the widest aperture if you’ll be working in low light (with ultra wides, depth of field is rarely an important point), but it’s a bit soft at f/2.8, so the aperture advantage isn’t huge (I usually shoot it at f/3.5 if I can to get it sharper). It has very little vignetting and distortion, probably the least of the group. Unfortunately it does show quite a bit of chromatic aberration at times. Overall it may be the best image quality of the group.
The Tokina 12-24 f/4 PRO is built like a sturdy tank (and therefore a bit heavier). Its a good lens but does tend to give low contrast images when shot into the sun and has a tendency to show some chromatic aberration when objects are backlit. This the one I’d take if conditions were rough: I pity the rock this bad boy falls on. Poor rock.
The Sigma 10-20 f/3.5 has a bit more distortion than the others but delivers very nice images and is also built well. It does everything well.
The Sigma 10-20 f/4-5.6 gives away some aperture to its more expensive brother. It is built about the same and the image quality, as best I can determine, is very close. It has the advantage of only needing a 77mm front filter as opposed to the 82mm filter of the f/3.5, so if you plan on using a circular polarizer (not always the best idea on an ultra-wide) you might prefer this one. Plus it’s the least expensive of the bunch.

But like I said above: they’re all excellent. We hardly ever get anything but happy comments about any of them.