I’ve offered this information a few times so I thought it would be sensible to put it all together in one list. Please do email or comment with suggestions, additions and alterations!
This is not an exhaustive list, just my favourites.

Berwick Street Soho is the home of the best shops:
Cloth House (1) for fancy silk, jersey, felt, wool, embroidered and decorated fabrics, smocked and gathered fabric, metallic, plastic, pvc, silk velvet, leather and all manner of amazing fabrics you have never seen before. I recently bought silk tulle and admired several types of silk jersey. I get my felt from here that I use for cushions.

Just across the road is Berwick St Cloth Shop, which is best for silk dupion and organza, as well as a huge range of embroidered, beaded and other fancy fabrics, as well as amazing silk taffetas and bridal silks, brocades and masses more.

There is a similar shop just up the road towards Oxford Street which just sells silks, and seems to have much the same range as the Berwick St Cloth Shop (No website – 020 7287 2881). Press on north and you will pass Noel Street. Monday-Friday you can stop here and visit Kleins, a haberdashery heaven. It’s a trade shop, so most of the stock is behind the counter, so ask for what you need and they will find it if they have it. I get bag handles and fasteners, as well as all the basics such as zips, thread, buttons, ribbons and corset hooks, boning, interfacing and masses more. Explore the website first to plan your visit, and even ring ahead if you are in a hurry.

Just past Noel Street is the second Cloth House which specialises in cottons and is amazing. I have seen organdie, corduroy, cotton velvets (in loads of fabulous colours), tie silks (I do mean silk for ties, not Thai..), Provencal prints, Indian prints, embroidered and appliqued cottons, Hawaiian style prints and masses more. Recently I saw a huge range of coloured Khadi and antique French linen. They also carry a wide range of vintage trimmings which are great. I love it. It’s probably the best place for quilters in London, though by no means is a patchwork fabric shop.

Just round the corner is Liberty of London which should need no introduction to fabric fans. Their cotton lawn is stunning quality and the prints range from trad to contemporary and new fabrics seem to come out regularly. I like to pick up small pieces for bag linings and used some to line a wedding suit commission. What is less well known is that Liberty have a great yarn department and nice helpful knitting staff. There is a knitting group in the Arts Bar Cafe on the (I think) 1st and 3rd Thursday of every month. I used to go regularly and they are lovely, and the cafe is great! The shop is by far my favourite place in London to browse fashion and home stuff and generally lust after stuff I can’t afford!