Best Mud For Backyard

Depends on what you mean by sinking. The typical answer here is mason sand is the best for leveling your yard. In general for a big area, I'd think top soil if you can find it. You want to avoid river silt as it is likely to carry weed seeds that you can't handle selectively (torpedo grass)

On a secondary note, is it a Slidell or New Orleans thing to call dirt/soil mud? My wife's family is from Slidell and they call it mud as well.

Just imagining a big pump truck and a guy squirting a massive amount of mud all over your yard