Of the available digital wideangle zooms, 10mm is the shortest minimum focal length (Canon and Sigma), while 12mm is the longest (Tokina), with the Tamron splitting the difference at 11mm. Note all of these are nominal numbers and measured focal lengths are rarely, if ever exactly as marked.
So just what is the difference between 10mm and 12mm at the wide end of a zoom? 2mm doesn't sound like much, but if you think of it as about a 20% change in focal length it sounds like a bit more. Below is a comparison of the framing of shots taken with the Canon EF-S 10-22 and the Tokina 12-24 (framed in red). Note that these are based on analysis of the coverage of actual images and the actual framing obtained from the two lenses, not a theoretically calculated difference. As such they reflect the true difference in focal lengths rather than any "nominal" marked focal lengths.
Though the difference at the long end of the range is also 2mm, it's now only a 10% change in focal length and makes a correspondingly smaller difference as shown below:
Below is a shot taken at 12mm at f4 using a standard thickness Tiffen circular polarizer. You can see very slight corner vignetting, most visible in the top right corner.
If you want no filter related vignetting at all, even wide open at 12mm, you would probably need to use a slim polarizer. However if you have a normal thickness polarizer, it's still quite usable if you don't quite zoom out all the way or if you're prepared to do a fairly east digital correction.
Without the polarizer vignetting isn't really noticeable at 24mm (above, left), but can be seen at 12mm (above, right). Stopping down by a stop significantly reduces vignetting, but even wide open it's not really very visible unless you are shooting scenes such as those above with a uniform sky or other uniform object covering the image corners.
As you can see, there is a small amount of barrel distortion at the wide end of the zoom range, but very little at the longer end. Distortion is rather easily corrected in digital images, so I don't see it as being a significant problem. It's small enough that unless you were shooting brick walls, you probably wouldn't notice it!