I have had shoulder rats that have gone litterally everywhere with me. This included outside. That being said, it really depends on a lot of factors the main ones are the area you are taking them, the other being the rat itself. Id never take a rat with me to say, a dog park, a grocery store, a place with a lot of natural predators I am aware of(Like, an area known to be a hunting ground for a bird of prey. Actually, where I would take certain rats depends on the rats themselves. McTavish, one of my past rats could go many different places with me. He sat on my shoulder, was very calm and not easily startled. One of our favorite places was the backyard, the library(NO not many people can get away with this. At this library I could.) which was quiet but busy with people, and the hardware store which was not very busy, but was louder than a lot of rats could tolerate. When I got Gaelan, he was more skiddish around loud noises, and did not shoulder sit as well. I kept him on a harness until he became more used to going around with me, but until he was over a year old the only outside places he could go was my yard, and one my shoulder walking down my small dirt road that had no traffic, ect. Otherwise he was not trustworthy. Now, however he has relaxed much more with age and I would trust him in my hood or pocket going many places. He is very laid back. Eros, I could never take anywhere. He is not a trustworthy shoulder sitter, though he too since getting older can go as far as maybe the stretch of grass in front of my building, provided he is easily accessable (in my hood where I could grab him fast, in my pocket with one hand one him, ect). I would not trust him elsewhere. Gallagher and Keiran after I adopted them I would not have trusted them anywhere outside with me, ever. They do not have the temperment that would allow for that. On the other hand Lochlan, a past rat of mine could go anywhere with me, and he did. He went to fairs, parades, stores, houses, park. He was a very special rat. I would never let any other rat go with me to those places. He was also partially crippled and should he have gotten away somehow(he would have never tried) I would have had no problems grabbing him. He was a special case. He went to a parade and the marchers handed him candy( :roll: which he accepted but didn't get to eat) and was loved on by a biker gang stopped in the traffic. I would never trust any other rat like that, ever. OUt of my current ratties, Gaelan could go some places outside with me, Eros none, Gallagher none, Rowan none, Seanan possibly eventually some but she is too young to be trustworthy enough yet. The other baby girls I have here, all are too young, active and playful to make any real guess.
So it depends on the rat, and depends on the places and depends on how well you have trained them to stay(train train train before you ever risk carrying them outside even on a leash!!), and how used to certain things(noises, strange people, ect) they are. You have to have 100% confidence that nothing will go wrong. Keep them on a harness is a great thing to do. Take them with you but dont let them run free, EVER.