I'm in the desert so we don't get really big mud puddles regularly. Most of the water we encounter is during our monsoon season and it's typically in the form of rushing water in rivers and washes. If obviously shallow and not moving fast we just cautiously drive through, otherwise we have to get out and check it out. This involves things like tossing decent sized rocks into the water and listening for either the "clunk" of hitting other rocks (shallow water) or a "glug" (deep water). We also regularly wade into the water to check for hidden deep holes, washouts, and to check the current. Even if the water isn't very deep, the currents can be brutal. Since our precipitation comes in the form of monsoons, most of the time there's water its because it has flash flooded, and is therefore moving pretty fast.
For example, here's
@1Louder trying not to get swept away while walking a crossing for us some time last year. We determined the current was much too strong and had to turn around at this point.