OK, I've researched this a bit more, so I'm going to have a go at answering my own questions.
A child born in Singapore to a Singaporean citizen and a British citizen (by birth) is automatically a Singapore citizen and a British citizen.
You don't have to register as a citizen in either case. You just are.
The UK is perfectly content for you to have dual citizenship. As a Singaporean, you have to pay attention to Article 135 of the Singapore Constitution, which states:
Deprivation of citizenship on exercise of rights of foreign nationals, etc.
135.—
(1) The Government may, by order, deprive a citizen of Singapore of his citizenship if the Government is satisfied that —
(a) he has, while of or over the age of 18 years, at any time after 6th April 1960 voluntarily claimed and exercised any rights (other than any rights in connection with the use of a passport) available to him under the law of any country outside Singapore being rights accorded exclusively to the citizens or nationals of that country;
(b) he has, while of or over the age of 18 years, at any time after 6th April 1960 applied to the authorities of a place outside Singapore for the issue or renewal of a passport or used a passport issued by such authorities as a travel document; or
(c) he is of or over the age of 18 years and has, whether before or after attaining the age of 18 years, been ordinarily resident outside Singapore for a continuous period of 10 years (including any period of residence outside Singapore before 2nd January 1986) and has not at any time —
(i) during that period or thereafter entered Singapore by virtue of a certificate of status or travel document issued by the competent authorities of Singapore; or
(ii) during that period been in the service of the Government or of an international organisation of which Singapore is a member or of such other body or organisation as the President may, by notification in the Gazette, designate.
(2) For the purposes of clause (1)(a), the exercise of a vote in any political election in a place outside Singapore shall be deemed to be the voluntary claim and exercise of a right available under the law of that place.
(3) Where the Government has made an order under this Article depriving a citizen of Singapore of his citizenship, he shall cease to be a citizen with effect from the date of the order.
(1)(b) and (1)(c) are pretty clear - in particular, don't get a British passport after the age of 18 or you risk being deprived of your Singaporean citizenship and being stuck as a Brit forever. However, I can imagine that (1)(a) is open to all sort of interpretation - I couldn't begin to tell you what rights I have exercised that are only available to a British citizen.
(This rather glosses over the point that the government has to actually make an order to deprive you of your Singapore citizenship. I wonder how many times it has actually done this?)
If I want something to prove that my Singaporean daughters with their Singaporean passports are British, I can have their passports stamped with a right of abode. I think I can also get a letter from the Home Office if I want to.
So, in a nutshell, dual citizenship *is* permitted in Singapore, and the likelihood is that my children will be both Singaporean and British for years to come. You just have to navigate around Article 135 as set out above.
