Speed tokens during opponents' turns can only be spent at the start of the turn, the end of the turn, or in response to being put over an Edge space. You don't get a chance to spend it during the action phase.
For reactive actions, like the ones granted by Chaos tokens or the Reversal Form, you get the opportunity to use those after the active player has performed an action, and only one reaction per action.
The step by step turn order can be found on page 28 of the PATCH'd UP update pdf.