A 3 blade stockman, a Barlow, or a small trapper. Any of these will do anything you need, given the requirements you listed. I've been using a 3 blade stockman to skin/dress small game for the past 25 years. The availability of spey, sheepsfoot and clip/California clip blades helps them handle any small game dressing chore you throw at it, save for joint breaking.
I've used a 1979 Uncle Henry 897UH for this purpose for a quarter century.
Currently, Case offers the best quality and most appropriate knife for the job.
A medium stockman, like my UH has a california clip blade (looks like the blades on the muskrat). and full sized stockman has a regular clip blade, and the blades are generally wider and thicker than a medium stockman.
Plus, you can get the Case in either stainless or carbon steel. Carbon would be my choice, though stainless would work just fine as it does with my UH.
For whittling/carving, carbon is head and shoulders above stainless. For dressing, stainless is nice.
A full sized trapper is too big for squirrel/rabbit/birds. It's designed more for critters like large raccoons, foxes, beavers, bobcats, coyotes, etc. The blades are twice as long as what's ideal for what you want to skin.
That's why I quit using my Case trapper for small game.
I bought it specifically for small game and found out that my medium stockman is much better suited to the task.
The california clip works well as a boning and sticking blade. The spey blade is nice for skinning since it doesn't tend to poke through the skin. The sheepfoot is nice when you need to apply some pressure to the tip when cutting.
Buy good quality. Buy a Case, if you can afford it. It's a knife that will last you a lifetime.
Don't buy a cop knife for small game. The cop knives are ill suited for that kind of work. It's like using a meat cleaver to fillet fish.
Spyderco and similar knives have their uses, but small game dressing isn't one of them.
Pocket clips on knives are a great way to lose knives. All the knives I've lost had pocket clips. They were all clipped to the pocket like they were intended to be. They all got knocked out of the pocket and lost. I no longer carry a knife with a pocket clip. i got tired of losing nice cutlery.
carry the knife in your front pants pocket, or in a pouch/sheath. Even sheaths tend to get caught on things and can be damaged. Pocket carry is the most secure carry for a pocketknife.
You could use a sodbuster jr, but the blade design isn't ideal for the task.
A good Barlow works well, too. A stockman works better. Either one should serve you well.
