I have seen both:
- "in the street" AND "on the street"
- "in the square" AND "on the square"
Which preposition is correct?
Prepositions are difficult: there is a lot of flexibility and cultural variation. These are loose guidelines for British english.
If you were talking about a person, in the street means that they are standing or walking in the street, Whereas on the street means that they are homeless. Likewise, in the square means that they are standing or walking in the square, and on the square means that they are honest.
For a vehicle or building you would use in for the street and the square and a named square, eg Times Square, and in or on for a named street.
If you are
in the street you are "within" the street. You are confined within the street.
If you are on the street you are talking about the street as a surface.
The same for square and other similar places, such as field, playground.
I'm in the square, field, playground = I'm confined within the square, field, playground.
I'm on the square, field, playground = I'm on the surface that is the square, field, playground.
Same for a large mode of transportation: if it's large enough inside to contain a surface you can use on to refer to the inside surface. Therefore, you can be both in (within) a bus, train, boat, airplane, elevator or on any of these.
"In the street" means literally that you are "in the middle of the street", just standing on it
"On the street" can be used to describe, for example, buildings, which face the street.
"The factory was on Jackson Street."
The same construction can be used in the case of describing a pedestrian who is walking down the street, but he/she is on the sidewalk, not in the middle of the road.