I usually (though not often these days) go for Dear Sir/Madam. If you use that greeting (or just Dear Sir) you should end with yours faithfully. If you start with Dear Mr (or Mrs) Bloggs you end with yours sincerely.
I remember this as - if there's an s in the greeting (Sir) there isn't in the ending.
I disagree - it is gender neutral, for one thing and that's a help. The fact that a form of address has been in use for a long time isn't automatically a reason to label it 'stuffy' or old-fashioned. If that was the case, nobody would use 'yours sincerely' yet it appears on millions of business letters every day.
'Dear Sirs' is the greeting that is old-fashioned and stuffy because it expresses an assumption - that all the recipients of business letters are male, when in fact at least 50% of them are probably female.
One of Britain’s most prestigious law firms has banned the use of “Dear sirs” from all of its legal documents and communications, apparently the first of the “magic circle” of top City legal companies to do so.
Freshfields Bruckhaus Deringer has stopped using the phrase from this weekend. In the UK the firm will now address all communications to “Dear Sir or Madam”