If you are looking to organise an event as a postgraduate student, the main option is to book the Postgraduate Common Room and Terrace on the 7th floor of the Graduate Centre, at the other end of the hall from the Reading Room. It can be booked for events focused on postgraduate students either during the day (i.e. before 6pm) via the Library or for events in the evening (after 6pm) via the Events team, with capacity for over 30 people. See the Library’s Graduate Centre page here.
When booking an event space through the Queen Mary Events team, there is usually no fee charged for the space on the condition that the event is catered by Queen Mary Catering, details here.
There are also QMSU spaces available to book for QMSU-affiliated groups, such as clubs and societies (but not for individual use by students). Events that are co-organised with reps can receive QMSU staff support and be hosted in SU spaces for free. More information is available here.