As far as I know, they allocate accommodation on a rolling basis - so they give out a set proportion of each type of room each week.
I applied around the 24th/25th March two years ago, and got an offer for my first choice (a single room at Passfield) within 24 hours.
Given that accommodation has been open since around February though, I would guess that around April the main LSE halls (so Bankside, Carr-Saunders, Rosebery, Passfield) might start to run out of rooms. If you can apply before the end of March, I would.