Is it possible to create at uCoz a forum on phpBB? Or to install vBulletin, IPB?
No, it is not possible to use PHP scripts and MySQL at uCoz. The majority (90%+) of free scripts are written by beginners in programming, who still learn to program, and so these scripts are unsafe for the server and therefore for other users’ sites.

We make up for the impossibility to install own scripts with many services, by means of which anyone can make a unique site of various levels of complexity. We do not plan to add the capability to install own scripts.