The biggest thing you need to worry about with regard to rain is the danger of a flash flood in a desert wash. Rain can happen a couple of miles away and if you are in the wrong place at the wrong time 20 minutes later you could get the surprise of your life. It is not something to scratch your plans over, but it is something you should be aware of. Crossing a wash is generally not high risk, especially if you can see a ways upstream. Driving for miles in a wash is another matter. And many desert trails do run up/down washes, sometimes for miles at a time. IF it is raining upstream then you don't want to be there.
There may be some mud, but most of the desert is rock. Most topsoil has been washed away over the years, leaving a rock top layer. In some cases there is soil under a layer of gravel, and what is there will most likely be clay, and thus good and sticky, but that is a minority percentage of what you will likely encounter.
Best bet is to study up on a particular trail. And if in a park, try to find the park headquarters and inquire about current trail conditions.
