First of all, at least in my view, comparing the Mosquito and the 22A-1 is a little bit of an "apples vs oranges" type of situation. One is a 22LR version of a “duty/carry” gun and the other more of an informal "target" gun. You can certainly plink, practice/train and shoot targets with both but their "style" is completely different. JMHO.
As far as the guns themselves are concerned, the Smith 22A-1 is a decent gun with lots of features. It's probably not going to be as long lasting as a Ruger Mark series but it doesn't cost as much either. Smith 22A’s shoot very well, have good, adjustable sights and, having a full length rail, are easy to mount optics on. I've had one ever since they first came out and am very satisfied with it. I would however recommend against shooting hyper velocity ammo through it as it is a bit hard on the slide/action. However, this is true for a lot of 22 semi autos that are out there. Some people make a big issue of the fact that the Smith has a nylon "buffer" in it that may need to be replaced periodically. It's purpose is to keep the slide from banging into the frame at full travel. The Browning Buckmark has a nylon buffer, as do lots of other 22 semi autos so it's not really a big deal. I shot my gun for years and never had the buffer break. Finally, I replaced it while I had it apart just because it was getting old, but kept it as an extra spare to have on hand just in case.
As for the Mosquito, the bottom line is that it seems like I’ve heard more complaints concerning function/reliability from Sig Mosquito owners than from the owners of all the other “quality” 22 pistols currently on the market today put together. If you want a “duty/carry style” 22 auto pistol I’d get the Ruger SR22, the Walther P22, or the Colt 1911 Government Model. I don’t know enough about the S&W, M&P 22 yet to comment on it. The Walther doesn’t have a de-cocker which I personally think every DA/SA hammer pistol should have. Walther, of all companies, should know better…. think P38. Also, although it doesn’t bother me, some people don’t like the trigger guard mag release. The Ruger SR22 works very well and has a de-cocker built right into the safety where it should be, unlike the Mosquito. Yea, I know, that’s the way Sigs are and they don’t have safeties…still… The Colt 1911, well, it’s 1911ish and I just find it to be the most fun to plink with of all the duty style 22 pistols……