I go mushroom hunting (a.k.a. hiking with stops to look for fungi) in very rainy/muddy conditions and use rain boots with wool socks for those really yucky days, and I insert my trusty Smartfeet insoles for support

In general, if it's just slightly muddy and damp (i.e. some showers here and there), I use my New Balance Gore-Tex trail runners -- lighter weight, pretty stylish, mud washes off easily and by and large, my feet stay dry unless I step into a pool of water, big time. I also have an ancient pair of North Face Gore-Tex trail running/hiking shoes that have detachable gaiters and these things are amazing -- I use them for snowshoeing and also sometimes in lieu of rain boots if I expect to get a lot of mileage in a day. I used to have Merrell hiking shoes too (Gore Tex, of course) and those were very good too.