If perchance you have some sort of aftermarket internals, then please DO seek the recommendation from the various menufacturers involved. For instance, I'm running a Wavetrac differential in my car that has a lifetime warranty --- unless I use a non-approved lube. For an open (non-limited slip) diff any good quality lube will work fine, either dino or synthetic as is your preference. I'd recommend using something that meets API GL-5 specs.
"API Category GL-5 designates the type of service characteristic of gears, particularly hypoids in automotive axles under high-speed and/or low-speed, high-torque conditions. Lubricants qualified under U.S. Military specification MIL-L-2105D (formerly MIL-L-2015C), MIL-PRF-2105E and SAE J2360 satisfy the requirements of the API GL-5 service designation."
Things get "sticky" when your talking limited-slip axles. My Wavetrac for example states explicitly do NOT add any "limited slip" additive or use a lube that states it's formulated for limited-slip axles because it works differently from a traditional clutch or cone type axle, and the additive will make it NOT work. - and will void the warranty.