School
The school that Lain attends along with her classmates Alice, Reika and Juri has no name in any of the works. Alice's teacher and Lain's teacher work here.
According to Visual Experiments Lain, it's called Ouka Private Girls' Academy (私立鷗華女子学園, shiritsu ouka joshi gakuen). It contains a middle school, high school, and college all on the same campus.
Description
The school buildings are mostly white. There is a main gate all the students enter through, beside which there is a bust of a man (unidentified, presumably the school founder). To the right of the entrance is a booth where the security guard sits.
In the classrooms, students are seated at simple individual desks facing the blackboard at the front of the classroom. There are large windows lining the outside wall and the wall facing the corridor.