I don't trap anymore. But in my younger days when I did I found a good 22 LR handgun and some times a 22 WMR was about as good as I could get.
I used a Colt Huntsman, a Ruger Standard auto, a Colt New Frontier 22/22mag a Smith & Wesson K22 and a Smith & Wesson M63 stainless kit gun. All were fine.
The one that was a favorite because I liked it's versatility and the ability to handload ammo of differing levels of power was a S&W 6" K-frame 32-20. It had all the power I needed to any handguns range (I could shoot pretty far out with good handguns in the day when my eyes were still good) For trapped animals I used a 100 grain bullet loaded to only about 750 fps. No hide damage. Loaded with the same bullet and a stiff load of 2400 I could get the bullet out fast, and it shot flat.
Looking back I am 100% sure I'd have been fine with a 38 special or a 9MM too, but I used 22s for the most part except when I could use the 32-20 which was my favorite, but that was a borrowed gun, so there were many times I didn't have it. I have one identical to it now, but now I just enjoy shooting it and carrying it as a back-up for a rifle. I think back to the days when I was a kit and had the S&W 32-20 along, and it makes me smile.