In general, where's a good place to start looking for muddy patches? And tips on avoiding getting seen?

Are you in the UK? In which case I'd say muddy estuaries are your most reliable bet. Initially scoped with an OS map (which can be turned on as a layer in Bing maps), then studied with Google earth or similar satellite view. Look for "saltings" - the marshy bits at the edges of the estuaries. The maps will also give you a reasonable idea how private its likely to be - somewhere close to a right of way (for your access), but not actually on one can be good. Look for accessible fresh water nearby for cleanup. And don't get stuck (its actually harder to get stuck than you might think, in my experience. I think most people who get stuck are trying to stay upright, and so sink in, whereas we're usually happy to lie down and wallow!)

Good question depends an where you around on the planet

However a few things if it's outdoors check the mud spot for rocks and tree branches

Look for water. Especially moving water. Mud is always nearby. The more water the more mud.