For SIA you don't need contract agency. It's direct application. Just a few other companies are having HR departments, big enough to deal whith the avalanche of applications or qualified people to do the selection. SIA,Cathay,Emirates,Etihad,Qatar,Gulf Air are among the companies you could look at. All the others,to best of my knowledge are using contract agencies. O,and Im talking about established carriers.
Good luck buddy!