My bilingual dictionary simply says the idiom means "to appear suddenly." No examples are offered. I wonder exactly in what situations this idiom can be used. Would you please give me some examples? Thanks.
'To appear suddenly and from no expected source', I would say:

'Just as I thought the highway was deserted, a policeman appeared out of the woodwork, pulled me over and gave me a speeding ticket.'

'I thought I would never be able to afford university, when out of the woodwork a lawyer contacted me to tell me that my long lost Uncle James had died and left me half a million dollars.'

The idiom comes, I believe, from the habits of the deathwatch beetle or similar woodboring Lamellicorn beetles which live as larvae within the woodwork of buildings, then emerge from the wood without sign or warning as adults.
Students: Are you brave enough to let our tutors analyse your pronunciation?
Thanks MM.

More than clear.
I now understand the idiom contains a sense of suddenness AND unexpectedness.
Can I say then it's synonymous with [(come/appear) out of the blue//out of nowhere//out of thin air]?
Yes, but I think 'out of the blue' is slightly luckier, Komountain. It would have worked better in my 'rich uncle' example.