When does an oscillating charged particle emit a photon as a result of acceleration? As far as I know a charged particle emits a photon when it undergoes acceleration.
When it comes to collisions it seems quite straightforward that a single photon is emitted with an energy related to the energy lost by the charged particle.
However, in case of an oscillating charge, or more generally, in case of a charge experiencing acceleration that is changing with time, it is unclear to me what is triggering the photon emission. Is it the change of acceleration? Is it something else?
 A: Classical electromagnetism describes this radiation well and is used for calculating the radiation.
Photons though are in the realm of quantum mechanics.  Quantum mechanical calculations introduce some corrections in synchrotron and bremsstrahlung radiation. 
For example bremsstrahlung , braking/decelerating  radiation can be calculated using the tools of quantum electrodynamics. Here is a Feynman diagram for bremsstrahlung 

If one calculates it one will get the probability distribution for a photon  to be emitted, and this probability distribution describes the spectrum . Individual photon emissions follow this probability distribution, as with all quantum mechanical interactions, individual photon measurements will look random, but they add up to the probability distribution predicted.
Corresponding diagrams can be written for acceleration, where the virtual photon does not come from the nucleus but from the  field in an accelerator, creating synchrotron radiation. Again individual photons will look random, but the cumulative distribution is predicted by the quantum mechanical solutions.
There is no "trigger" for the photon emission other than the exchange of a virtual photon that will generate the probability functions for a real photon to appear. 
