Start at the bottom of the page
Read the foot of your sheet before you read the top of it. The slip the Legalisation Office attaches speaks for the signature and the seal already sitting there, which makes the first question a simple one: who signed this? If the answer is an officer of a public body, the paper is on the right track. If nobody signed it, there is nothing yet to mark.
Where a usable signature comes from
For civil records the name belongs to a registrar, and you never have to risk the certificate you already own. Westminster Register Office at 317 Harrow Road, London W9 3RJ issues a fresh certified copy drawn from the same register, and that copy carries a signature of its own. Send that certified copy and keep your own certificate at home.
Other papers pick up their signature elsewhere. A court seals its own orders, a company's officers sign its filings, a school signs a record of results. None of those bodies issue the mark, and none of them need to be told where the paper is going. Their only job is to put a name on the sheet that the British office can act on.
Why a second stamp follows
Syria has not joined the Hague Apostille Convention, so document legalisation for Syria does not end with the British slip. An apostille from United Kingdom to Syria is followed by a stamp from Syria's consular staff in Britain, and their charge and their pace belong to their side of the job. The British half is the published, predictable half of the route.
What to have ready
So what you need for an apostille for Syria at the British stage is one signed public version of each paper and £45 for each of them by post, or £40 each on the next-day line if you are a registered business. The office works through PO Box 6255, Milton Keynes MK10 1XX and sends the marked sheets back to you.
One last look before the envelope is sealed. The name on the signature line should be the full, current one the issuing body puts on that record, rather than a mark copied onto a form by hand. Where there is real doubt about it, ordering the paper again from the body that holds it costs far less than a bundle that comes home unmarked.