"I am starting to take my dog into stores and other public places. How do I handle him having a bowel movement on the floor? He is house trained. He is small breed and 1 year old."
Good for you for asking before you start! It can prevent headaches for you, your dog and the places you visit!
The key thing is to remember that dogs don't generalize well. That means that even if he's housetrained at your house, he may not understand that he's not supposed to go in other buildings. He may also go due to stress (excitement or worry). Urine production increases when a dog is stressed. It's the same as us humans. Ever needed to "go" more often when you are excited, stressed or cold?
The key is to put potty on cue and teach him to go just before you take him in. Find a spot in the parking lot just outside the door. That way you are pretty sure he isn't likely to go inside.
Keep training sessions short inside businesses until you know he is comfortable and he learns to potty only outdoors. Take frequent breaks outdoors.
Then, if he does go, it will most likely be due to stress (excitement or worry). That's good information to slow down on your public training and give him more acclimation time and keep sessions shorter. Other reasons are he might have eaten something that didn't agree with him. That's a good reason to keep working on leave it and make sure other people at home aren't feeding him junk treats.
Be prepared for it by carrying plastic bags, paper towels and wet wipes so you can clean it up. Just be matter of fact and try not to be embarrassed. It happens to everyone at some point! It also helps to alert a staff member so they can disinfect the floor afterward.
Keep an eye on boys to make sure they don't lift their leg on merchandise. Simply interrupt the behaviour with a Kissy sound and keep him moving to the outdoors. Cue the potty cue once you get to an appropriate spot.