Removing marks from ceramics

It just depends what substance has caused the damage. As Naomi said, any kind of oil/fat/grease is a lost cause, as I've found with many of those large blue and white plates. Another lost cause are earthenwares that held spirits or liqueurs, like those
Wade barrels and the Drioli bottles. Earthenware oil and vinegar bottles can be similar.

Having said all that, (thin) bleach is great, particularly on things like Poole or any
pottery that crazes badly, like Portmeirion, Midwinter etc.
Immerse in a plastic bath, sometimes for a couple of weeks, to let it really soak in,
then warm up with a gentle heat to burn it off, and get rid of the smell. Rinse off any
crystals and wash & heat again.
Bleach also gives almost instant results on stained teapots.
2 litre bottles are literally a few pence from all major supermarkets.

Strong Peroxide, as used by the hairdressing trade was really the best as it was
odourless and would shift anything, but I don't think it is available anymore due
to legislation.

I do use wood bleach (Oxalic acid) on furniture, though I've never tried it on pot
as it's pricey and only comes in tiny bottles, but would probably work aswell.

Metal marks on tableware, or those TG Green mixing bowls, are easily removed
with liquid metal polish.

Cheap bathroom cream cleaner will get most surface mucK off, eg. foot rims.

One downside of cleaning is of course damage being revealed that you had
missed when you bought it!

Wasn't he the infamous slum landlord?

Of course I should add the rider, that many items of pottery should not be cleaned
if they are painted or enamelled or also any Gilding/Silvering or damage will result.
This would include a lot of 1920s/30s pottery like Clarice/Susie Cooper/Grays/
Charlotte Rhead/Bretby/Brentleigh etc. where the decoration is unglazed.