I see there's been lots of discussion about mains transformers and coupling. The key factor is CPS (capacitance between primary and secondary). A standard foil electrostatic screen between primary and secondary on an EI transformer will reduce CPS from 250pF on a 50VA transformer to around 5pF, perhaps 2pF. Adding capacitance to chassis from each end of the secondary creates a potential divider to common mode interference. Entirely covering the secondary with copper tape (without creating a shorted turn) reduces CPS to about 0.1pF. It's fiddly work and requires a split bobbin transformer. Topaz Ultra-Isolator transformers fitted individual foil electrostatic screens around primary and secondary to achieve <10fF. They also have a safety screening plate between the two windings capable of sinking enough earth current to blow a fuse quickly. That's what it's there for; safety, not screening. Adding a safety screen to an EI transformer kit plus totally enclosing foil E/S screen on secondary, I achieved CPS = 8fF +/-60% uncertainty. But what I really learned was that it's terribly easy to squander the improvement of a good transformer by sloppy external wiring. If you do something to prevent mains common mode interference getting in (low CPS transformer, mains filter) you have to totally screen incoming mains. Folded tin plate screens work well over incoming mains wiring, with self-adhesive copper tape (gardener's slug tape) over all joints. RF crawls through the tiniest gap.
As has been mentioned, split bobbin transformers have high leakage inductance, needing an RC snubber across the secondary to damp their resonance when the diodes switch. Capacitors across diodes in the rectifier simply couple interference from the mains to your sensitive electronics. I know that's how it used to be done, but it was wrong. The Mark Johnson Quasimodo circuit works very well for determining the optimum snubber.