Well now.....here I had a very grubby 1950s Waltham Melamine dial. I read everything on this site and the overall view was leave well alone but oh no.....not me!, I had in my drawer some "Greygate" plastic polish which I use to shine up acrylic crystals so I thought what the heck! and proceeded to apply a little on a foam rubber make up removal stick in circular motion on the dial. Within a few minutes the dial was gleaming with all the ingrained dirt and staining removed, so a rinse under the tap and hey presto!.
I have used this type of polish before and what it seems to do is very finely cut the surface back to reveal new. Care has to be taken to only apply light pressure and constantly check, but with a little gentle massaging it has given me great results.
This polish is avilable on e-bay.
Bye for now.........

That's interesting Stephen. It must be a very gentle polish, as I'd normally be be concerned about damaging the graphics, which are just painted on the melamine dials.

My sentiments entirely Ed, I certainly proceeded with great caution, but this really is a very very fine polish so it is quite easy to control how much you do, it also seems to contain some sort of cleaning agent but who knows?.

Interesting solution!
In the past, I have not had the courage to use any chemicals.

I have used as recommended elsewhere, the idea of using a soft clean pencil eraser(rubber). This actually works quite well and does not remove the numerals if done with care.

But now you have given me a new thing to try.