Good suggestions all around, Mark. Much depends on your focus, but both the PRO and Colindale will yield rich finds for the crew, passenger and technical aspects of the disaster. The NMM tends to produce results for crew and technical interests. Both the PRO and NMM have MN and RN (also RNR) material.
Another one I'd add to the list for anyone focused on biographical details of passengers is the Family Records Centre (FRC) at Farringdon. Some of the material here - e.g. indexes births, deaths and marriage and census material, can be found elsewhere on microfiche or film, but it's a handy location to find all of it plus Wills and other such data. It's a very good place to start constructing a biographical background for any British passengers or crew you're interested in.
Another English archive that's worth a visit if you're interested in the Merchant Service is the Guildhall. Outside of London, many local records offices have good material. I've found the Liverpool Records Office to be an absolute gem of an archive, and made a couple of unexpected finds there. The Merseyside Maritime Museum is also useful. Outside of the UK most of my correspondence has been with local or national libraries (NSW's Mitchell Library was a good source in Oz, for example). A word should be put in for Newfoundland's Maritime Archives - this is a superlative source, with a very good staff. These are some of the folks that go above and beyond the call of duty. When a recent package of material went astray they went to great lengths to track it down, and finally re-copied and sent it registered mail. Of course, Mark and I are in England, so our research material biases tend to lean that way. I think you've also made use of the Ulster Folk and Transport Museum, haven't you, Mark? There's also the PRONI (Public Records Office of Northern Ireland), and I know Senan Molony has made good use of the Irish Republic's national archives. American based researchers have readier access to, say, the Library of Congress material etc.
Side note to Mark - have you been out to Kew lately and seen what the PRO is now? In April, it merged with the Historical Manuscripts Commission to form a new body: The National Archives. So technically it's now called 'The National Archives: PRO' or 'The National Archives: HMC'. The building out there still has 'Public Records Office' in that sandstone slab over the door, but everywhere else the PRO logo has been replaced with the NA logo. On their website, it retains the 'Public Records Office' with a subtitle 'The National Archives'. Think I'll just stick to 'PRO'...
