I am wondering what the best software is to design Minecraft mob textures.
Whatever editing software you're comfortable using is the best software to use. Ideally something geared towards low-res or pixel art, but really just about anything can work.
If you'd like some freeware recommendations, I suggest either Paint.net or GIMP. If you know how to compile the code, you can also get Aseprite for free. If not, there's also a fairly inexpensive paid version. (And yes, the compiling method is officially endorsed.)