There are a lot of private landlords, but you can not know what to aspect. Some of the landlords you can’t trust, so be careful with this. Especially when you come from outside Belgium and can’t visit your apartment before, you should make good appointments and make use of a contract
If you want to optimize your travel to Brussels, it’s not a bad idea to contact an agency or even a placement agency, they can even give you advantages like discounts for musea, etc. The second one even does all the footwork for you if you are to busy.
Beware of the fees the ask, they can differ and so do the types of services.

Trainees in brussels
accommodation for1 to 10 months for trainees in brussels, options include flat-shares, furnished studio flats and rooms in a local resident´s house.
rue Kindermans 9
1050 Brussels
tel.026462624
www.traineesinbrussels.be

Another type of accommodation which isn’t mentioned yet are rooms in guesthouses. 

Bed & Breakfast
Taxistop
rue du Fossé-aux-loups 28
1000 Bruxelles
tel.070222292
e-mail: bnb@taxistop.be
www.taxistop.be

Bed & Brussels
rue Kindermans 9
1050 Bruxelles
tel.026460737
e-mail: info@bnb-brussels.be
www.bnb-brussels.be