How is flame stabilisation carried out?
As usual, it is a combination of measures:


Flow speed at the start of the combustion chamber is the lowest in the whole engine, in the order of 30 m/s.
High fuel pressure and many small fuel injector nozzles, such that the mean fuel droplet size is kept as small as possible
Flame holders cause a pocket of stagnated, burning flow which helps to ignite the fuel-air mixture
The combustion chamber is long enough to allow most of the combustion to happen inside, such that radiated heat from the burning gas ignites the fuel-air mixture upstream.
In marginal conditions the ignitors are switched on to help speed up the ignition.
Control of parameters, such that fuel starvation by rapid throttle changes can be avoided. Examples are fuel control units, where a centrifugal feedback valve would control fuel flow depending on the engine's rotational speed, or electronic control (FADEC).


Much can be done by running the engine such that the risk of flameout is minimized. The operational measures to ensure a stable flame would be:


Avoid fuel starvation (empty tank, fuel pump failure, clogged pipes). This one is obvious, but I need to include it for the nitpickers.
Avoid too low pressure for rapid combustion: Pressure speeds up the chemical reaction, and if the aircraft flies high and/or throttles down, the lowered pressure in the combustion chamber might cause a flame-out. High-altitude jet engines have longer combustion chambers as a precaution.
A compressor surge or disturbed inlet air flow can cause a sudden pressure loss in the combustion chamber and a consequent flame-out, the latter especially in supersonic fighters with their long intakes and high angle-of-attack capabilities.
Avoid sudden cooling of the air inside the engine, e.g. by heavy rain. The additional water extracts energy when it evaporates and causes a temperature drop in the combustion chamber which can lead to a flame-out of a throttled-down engine. Another possibility is too much fuel which has been ingested by the intake in a botched aerial refueling attempt.
Avoid clogging of the inside of the engine by ingested dust, like gun smoke or volcanic ash.