For years I've thought it was virtually impossible, or prohibitively expensive, to get rid of black mould from around baths, and from tile grout. It is in fact, it's a piece of piss. All it takes is bleach. Specifically bleach that contains sodium hydroxide and sodium hypochlorite. Simply spray on, using a cheapo spray bottle, and leave, periodically checking to see if it's cleared. Occasionally it'll need a few applications. Then rinse off with fresh water. The mould actually disappears. Doesn't run off into a puddle of mould or anything, it just disappears. It's fuckin' magic.