Different color clays have different properties. Some dry out easier than others. Some are more brick-like when dry than others, etc.
Texas has a black clay the locals call "gumbo". It has the odd property of shrinking a LOT when dry. Texas soil will show very big cracks during really dry summers. Houses have to be built differently, and any pipes laid in the ground are installed differently, to keep this problem from breaking foundations and plumbing. It is still fun to play in, if you can find it wet and without ants.
Arkansas has lots of red clay, usually with sand mixed in. (again, I'm typing from experience, I have not done any sort of detailed survey) This is really fun to play in when wet, and re-wets pretty quickly. Maybe it's because I grew up here, but it's my favorite stuff to play in. The Red River has this sort of clay.
Out in Arizona, the Little Colorado River has a mix of red clay and sand, but it is the place I have found the most of what I would call "textbook" quicksand. It was the first place I ever walked along the bank of a river and found myself walking as if on a waterbed. I could see wave action almost 4 to 6 feet in every direction, but with the top being sturdy enough that I felt I could have just kept walking and not sink. (Of course I stopped to play, though!) My favorite bit of mud to play in used to be a mile away from the local city, which has expanded to both sides of the river, now.
The mud may still be there, but the privacy is lost.
Now, if you are serious about digging a pit, you will need something to do with what dirt comes out. This leaves you with a question: Are you going to buy something to go into the pit, or are you going to backfill with the dirt (minus rocks and roots) that came out of it. It is best to plan this ahead of time, as you will not kill a lot of your grass by making a big pile of dirt on it. If you are going to back-fill, then you will need a wheelbarrow to move the dirt over to your patio or some place without grass. You can't carry it one shovel at a time, as that will cause so many trips that you'll kill grass from walking. If you plan on back-filling with the soil that comes out, you still need to set the soil aside where there isn't grass, but you'll need a separate place to put rocks & roots.
If you do not plan to put the soil back into the pit that is coming out, and you have a big enough yard, just spread it out in the yard and water it. Water will take the new soil down toward the roots of your grass. If you do not currently have good grass, you can move the removed soil to low spots in your yard to level it out, or just spread it out if your yard is already level. They also make landscaping bricks such that you can stack them and put soil in for a flower garden, If you like. If you find you have rocks, roots and soil, but want to put something new into the pit, you can always ask your neighbors if they need any of what is coming out. If you're in the city, there may be some day of the week the city will pick up waste wood such as roots. They may also pick up your rocks, if you ask them. I piled a lot of broken rock out near the street here and the city sent a work crew to gather them to throw into a wash, down by the local park.
Probably more than you care to read, but I have done some digging, as you may have guessed. I did not plan ahead, and I have 1 unsightly pile of clay mixed with big rocks, one pile of just rocks, and a large pile of gravel. These would be gone already, but my wheelbarrow broke and I had knee problems back during the summer, so it's a chore that waits for better weather to fix these piles. I guess I'd rather others did not get this problem. (I don't even have a mudpit to show for this, either, I dug a French drain!)