I have a few areas of black staining under varnish where water has seeped in. Does anyone have any ideas / favorite method of removing it after sanding down?
Your chandler will stock a number of standard boat cleaning products for stained hardwood (e.g. Starbrite), that contain oxalic acid or phosphoric/oxalic acid mixes. Slightly cheaper alternatives are now available from garden centers. If the stains are extensive a much cheaper option is buy oxalic acid crystals - said to be available from chemists (though not round my way).
The black stains are the first signs of wet rot. This treatment is only suitable for hardwoods which resist rot, softwoods ought to be replaced. The cleaner will not totally eliminate signs of the mark, and some sanding back will be needed. Try to eliminate the source of the damp.