How do amnesties for illegal immigrants work with time conditions?

Here in the UK the Liberal Democrat leader Nick Clegg is proposing an amnesty for illegal immigrants who can prove they have been living in the country for at least 10 years, among other criteria. He argues the benefit of this is that it will bring illegal immigrants out of the shadow economy, so that they can start paying taxes and so on instead.

How does this work? How can someone prove they have been living in the country illegally for 10 years? In that video I just linked to Nick Clegg says that the current Labour government has been operating a 14-year “residency rule” which is effectively the same policy. Anyone know how that operates at the moment?

Are these time conditions an element of any proposed or enacted immigration amnesties in the US?

It’s probably easier in GB as it’s an island, so unless you’re stowing away on a barge, you are probably an illegal due to overstaying, rather than fence-hopping as is the modal form for illegal immigration in the US.

So there will presumably be a record somewhere of their entry (or a visa in an old passport). Apart from this, it’s not like illegals don’t have leases, buy cars, do other random things which leave paper trails. Also, they may accept affidavits from community members who have known the illegals for the requisite time period (religious leaders, in particular, are what i would have in mind)