When can I activate the special Halicarnassus effect?
The rules do say it (English base game rules, page 8 and also page 9, for the wonder The Mausoleum of Halicarnassus):


  Clarification: this special ability takes place at the end of the turn in which
  this stage is built. If players discard cards this turn (as in the case of the 6th
  turn of an age), the player can also choose from among them.


So when you build this wonder stage, you wait until everyone finished their action, and then you go through the discard pile and build one of its cards for free. One time only (per wonder stage).

Special cases:


if others use the same ability in the same turn (tl;dr: there is a defined resolution order)
if the Babylon (B) player discards the 7th card for money (tl;dr: it gets added to the discard pile before the Halicarnassus player goes through it)