How to filter out the mains power supply spikes?
Your 12V linear regulator is most likely running out of headroom, due to the fact that you have a fixed turns-ratio step-down transformer as its input source (which is of course gets rectified and smoothed). As the AC voltage coming in drops, so must the AC voltage coming out of it. Once the voltage gets too low, your linear regulators don't stand a chance.

You can either replace the whole power supply with a switcher as the other answer suggested, or use a switcher to generate a regulated 15VDC bus to feed the regulators  (instead of the transformer and rectifier scheme). If your loads are noise sensitive, keeping the linear regulators may give better performance (at the cost of lower efficiency).