IME they can make a noticeable difference, but it is somewhat system dependent, and mains quality/house wiring dependent. Also, some are better designed than others. I've extensively used the Russ Andrews woven ones, and, in my case, at least as much benefit came from following his (excellent, and free) tips regarding making sure the general house wiring/earthing are all in good shape, checking for loose connections, keeping contacts clean, using a dedicated low impedance ring-main, etc., etc.
I still use the ones I already have on my sources, etc., but haven't bought 3m pair runs for my DSPs yet, as my funds are still recovering from Meridian purchases

At the moment I'm using a good quality, heavy gauge mains cable (nothing fancy, just as meaty as possible) and using WattGate connectors both ends (which are excellent, heavy gauge contacts and plated too, etc.) along with Euro sockets to get rid of fuses, and it works well. That said, if I get an injection of funds, I personally would buy some RA powerkords, having experienced the difference before.
In terms of the points raised about lack of measurements, and proof, etc. on the original link, the RA vs ASA case was, in fact, quite interesting, as they
did go and measure the effect the cables were having - (
RA Test Results). IMO the ASA acted inappropriately the second time around, more like a witch hunt than an honest, normal investigation (which doesn't surprise me after witnessing the huge flame wars which seem to erupt on certain forums - not here, I might add - when this subject is broached). An interesting discussion about this can be found here -
WHF Forum Thread.
IIRC, in terms of the measurements, the particular mains cable in question, a woven RA/Kimber designed cable, in various AWG and braid patterns rejected an increasing amount of RFI in the MHz band. However, it had a 'comb' type pattern to it, and that was the basis of the newer 'sawyer disc' design, I believe, which attempted to smooth that out and provide a better, more broadband rejection, although I haven't tried that design.