I think your three gun classifications are roughly about the same as ours. Although I'm not an expert. The restrictions vary by state (like yankee.fan not being able to use a rifle in CT)....
For instance there are limits to how many shots you can have in your clip, etc.
In most states I think assult weapons are legal. However to have a concealed gun (pistol, snub noses) you usually have to at least register it.
Of course if you are a clueless person carrying "your husband's" pistol on an airline these days, you get a bonus price of getting your picture and story on the front page of CNN.com!